If we're not specifically instructed to do something, can we?

The principle that nothing is allowable unless it is explicitly allowed in the Bible is a bad one. This is also often referred to as "being silent where the Bible is silent", and is usually applied selectively to rationalize traditionalism (and traditionalism not tolerated by Jesus - see Mark 7).

This is best demonstrated by considering that Hymnals (like instruments) are not explicitly allowed in the New Testament, yet no proponents of this principle use it against Hymnals. Rather, proponents seem to use it to subjectively target practices that do not fall into their traditional comfort zones.

Furthermore, the principle is self contradictory, as it has no basis in the Bible. So it is necessary to abandon the principle in order to use it.

There is an aspect to this type of question that is dangerous, although surely this is not what you have intended. C. S. Lewis commented upon this in his article "Man or Rabbit?" in the anthology God in the Dock. The gist of his argument is that our heart is not in the right place if we are trying to see what we can get away with, as opposed to constantly struggling to find the best way to serve Christ, and in this instance to do the liturgy.

So perhaps the answer is that if you are not expressly told not to do something, but are not sure whether it is good to do so or not, then ask around (as you are doing here), pray about it, search the Scriptures, and, of course, see what everybody has done in the past, and try to apply all that to the situation. But there is rarely a pat or obvious answer in making decisions that haven't been made before.

In this instance (instrumentation in worship), tradition (in the Church of Christ and in my church) has seen fit to bar instruments from liturgical worship, "because it seemed good to the Holy Spirit and to us," I suppose. Did you want an answer to this, or to the more general question?

To add to what Tyles Gillies says, and in contradiction of your school's teaching, we are instructed to praise God with instruments. It's worth quoting in full:

Psalm 150

Praise the LORD.

Praise God in his sanctuary;

praise him in his mighty heavens.

Praise him for his acts of power;

praise him for his surpassing greatness.

Praise him with the sounding of the trumpet,

praise him with the harp and lyre,

praise him with timbrel and dancing,

praise him with the strings and pipe,

praise him with the clash of cymbals,

praise him with resounding cymbals.

Let everything that has breath praise the LORD.

Praise the LORD.

I especially love the idea of people taking the clashing and resounding cymbals literally. =:-)

Background

Silence of Scripture

This has to do with what we call the "silence" of scripture — things that are not explicitly discussed — and whether silence is permissive or prohibitive. In the general case (not just scripture), we often think of silence as permissive. However, it is easy to show that this is not always true. I'll provide an example:

Let's say you send your teenage child to the store for milk and eggs, and give him/her a $20 bill to cover the expense. The child comes back with milk, eggs, ice cream, and less change than you hoped for. You didn't tell him/her not to buy the ice cream, but you're probably not going to be happy. Here, the silence was prohibitive. You also didn't specify which store to go to. You may have expected the local grocery store, but if the child knew of a sale at a nearer convenience store they may have chosen that instead. Here, the silence was permissive.

Applied to scripture, there is a common (not universal) interpretation that when we are given instructions in an area of worship, silence on other aspects of the area is prohibitive. For example, God told Noah to use gopher wood when building the ark. He didn't mention other kinds of wood (he was silent), but his specific instruction for gopher wood excludes/prohibited them. What if Noah wanted to use gopher wood for the hull, but thought a little bit of maple trim might have looked nice around the window and door? This interpretation system would not allow for that. Going back to the shopping example, ice cream was not on the list, and so was prohibited. On the other hand, scripture says nothing about air conditioning our homes, or heating/cooling them at all, and so the silence is permissive.

Hermeneutics

The above system is an example of a hermeneutic. Hermeneutics are the lens you use to study scripture. As an example, they are how you know what to do when you come upon "Do not judge, or you too will be judged." in one place, and "Do you not know that the Lord’s people will judge the world?" in another. Hermeneutics help ensure that you interpret scripture consistently; that you not hear only what you want to hear from the passages you want to read. A perfect hermeneutic, perfectly applied, would lead to a perfect understanding of scripture. Unfortunately, man is far from perfect.

Most good hermeneutics will at some point need to deal with the silence of scripture, in some way or another. Most fail to handle it adequately. The hermeneutic described above actually handles the subject pretty well, but hermeneutics are a creation of man, and so are not flawless or perfect. Our application of them also often leaves something to be desired. In spite of that, if you don't have a hermeneutic system to use when you study scripture, you're probably not getting what you should out of your study. At the same time, it's important to remember that rather than hard-fast rules they should be seen more as guidelines.

Answering the Title Question

This specific hermeneutic is normally only used to apply to worship. Silence of scripture is more often taken to be permissive in the general sense, but for worship specifically it might be a good idea to pay more attention to what is and what is not enumerated explicitly. Therefore, with regards to the title question, the answer is most often permissive. However, it's still important to make sure that whatever you do, you're making choices that are centered on God, and not choices that are centered on what you can get away with.

Instrumental Music

Restoration Movement

To fully understand the instrumental music in the churches of Christ, you need to know something of our history. The churches of Christ descend from the Restoration movement of the early/mid-1800s. This was a movement that strove to throw off all of the excess structure that had arisen over the years since the 1st century and restore 1st century-style worship. The primary means of achieving this goal was through a very careful and strict interpretation of scripture. "Innovations" from known historical patterns of the first century should only be permitted when there is a clear case for them, shown either through direct command, example, or necessary inference.

The reason for the return to 1st century worship was a reaction to all the divisions, and a desire to fulfill the Lord's prayer "that they may all be one." It was a drive for unity, and one part of this was to promote hermeneutics that would push away anything that a true seeker might possibly see as contrary to scripture, and hold up only those things that meet an exceptionally strict criteria for doctrinal purity (the other two parts are congregational independance/freedom and an strong emphasis on individual study). Don't knock the process too hard, as it met with some success... the Restoration Movement was one of the few major groups to begin it's history as the result of a merger of a few smaller groups, rather than as a split from a larger group, and was the only American group in existence at the time to make it through the Civil War without dividing over the issue.

churches of Christ

There are four pieces that combine to contribute the general (not universal, but nearly so) use of a cappella music among church of Christ congregations.

Silence of Scripture

The first is a very strict interpretation on the silence of scripture, as described in the section above. The upshot is that churches of Christ believe we are lacking an explicit instruction to use instruments (while singing or otherwise) in congregational worship — that scripture is silent here. We discount the passages in the Psalms quoted in other answers as allegorical, specific to Old Testament worship, or applying to personal life rather than congregational worship. Churches of Christ acknowledge that we do have explicit instructions to sing. Because we have an explicit instruction about how to use music in worship, the silence in this case is prohibitive rather than permissive.

To put it another way, churches of Christ don't use incense during worship either, and you likely don't think this is strange — in fact, you would likely feel a little odd to see/smell it — after all, God does not instruct us anywhere that he wants incense as part of our worship. He's silent about incense, and so we choose to focus on the areas where He is more direct in what He wants. This argument is weak when left by itself, but remember that it's only part of the reasoning.

Practical Participation

The second reason is participatory. The command to sing praises is interpreted to mean that music in worship should include everyone as a participant... that if the music portion of your service consists mainly of just a few performers or even a choir, while the majority of the congregation is only listening (even if this is the effect rather than the intent), you're doing it wrong. The entire congregation should take part in producing the music. Instruments are seen as counter-productive to this effort, as history shows they have a tendency to take over and dominate the performance to the exclusion, rather than inclusion, of audience participation. Additionally, successful use of a cappella music in worship is something that takes a commitment. If you decide to just try out a cappella music one week in service, you're likely to fail miserably if few of the participants have done this before. Therefore, as a practical matter the choice to use a cappella music came down to an "all or nothing" approach, where those that use a cappella music in worship tend to do so exclusively.

Historical Record

The third reason is historical. Remember that this group comes out of a movement whose goals were to restore 1st century worship styles. We do know from historical documents (sorry, no reference handy :( ) that the earliest Christians, without exception, did not use instruments in their worship. At all. In fact, it's already the third century before even the most sympathetic scholars find any historical examples of any instrumental accompaniment. It's possible there were forces other than doctrine driving this: for example, funds, fear of persecution if louder instruments gave away a secret service, or simply modeling congregational worship after Jewish synagogue worship (which also did not use instruments). But the fact of the matter is that the historical record indicates a cappella is the way to go.

Identity

One other reason is that as a capella music for worship became less and less common in the culture, retaining and featuring a capella became part of the identity of the churches of Christ. Identity is especially important for a group as loosely associated as the churches of Christ, who recognize no earthly central authority or association. This final piece is by far the weakest from a persuasive or justification standpoint, but it still needs to be mentioned.

Application

Put the four together, and you have a fairly strong case that a cappella music is the way to go in the worship service. Unfortunately, more recent history in churches of Christ have forgotten that the reasoning here is as much practical as it is scriptural (especially with regards to the fourth reason, though, again, the scriptural argument is not entirely without merit) leading to a stronger reaction against instrumental music from some contingents within the churches of Christ than may be warranted. Remember, again, that this is a group with no earthly central authority or governing body, and therefore anything you hear (including this very text) is the opinion of the speaker; never of the churches of Christ as a whole.

There are also indications that you do not need to avoid instrumental music in personal and corporate worship. For example, the historical evidence is that the early worship followed the Jewish synagogue model. Jewish synagogue worship was entirely a capella, but they were happy to use instruments in their personal and family worship as well as the temple/corporate setting.

In other words, there is something to this silence of Scripture thing, but it's not fool proof. There's also something to be said for using a capella music during the formal church service. But you're probably in safe water to listen to a little Jars of Clay, Newsboys, or Reliant K with your friends or on your own time. The important thing is the make sure you are tuning into what God wants here rather than what you want.

There's more info on the subject, include a number of researched references into the history and scripture, at this link.

Regulations are created for fallen mankind who are unable to live by Spirit and truth. Furthermore, such regulations tend to be best expressed as "thou shalt not" and lend themselves to an end result of legislating every tiny little nuance. Just look at both the Hebrew law and the US law by way of example. That's because these laws are attempting to control behavior of a society that by and large is motivated to find the loopholes. We live with the attitude of, if it's not specifically forbidden I can do it. This can lead to erroneously attempting to counter this attitude by insisting on only allowing that which has been specifically legislated for, which leads to just the same problem from the opposite direction - now everything which is allowed must be spelled out.

But God's word attempts neither of these devices. The problem is that fully living up in spirit of God's law is much harder and more demanding that adhering to a bunch of regulations and rules (whether permissive or prohibitive, and no matter long winded the rule book is). I mean harder in that there is no set of boxes you can tick to tell yourself - "yep, nailed it". When I drive to work in the morning, on any given day I fully adhere to the state's driving statutes; but with every muttering under my breath and every irritated thought towards another driver I fail miserably at living up to God's perfect standard. So we legislate and regulate, in large part because it's comforting to have a standard against which we can be objectively measured and not found wanting.

Jesus, however, summed up the entire Hebrew law and prophets in two profound instructions:

Mat 22:37-40

³⁷ Jesus replied: “‘Love the Lord your God with all your heart and with all your soul and with all your mind.’ ³⁸ This is the first and greatest commandment. ³⁹ And the second is like it: ‘Love your neighbor as yourself.’ ⁴⁰ All the Law and the Prophets hang on these two commandments.”

Jesus is not after conformity to a bunch of regulations and rules, but a conformity of principles and attitude of the heart. Were it not for our brokenness, we would not need anything more than these two commandments.

Jesus "flaunts" a number of the laws that the Jews had on the books specifically, it seems, to make this point. For example, healing people on the Sabbath because "The Sabbath was made for man, not man for the Sabbath" (Mark 2:27).

Paul weighs in on this idea of personal conscience and accountability when he addresses the Corinthians on the matter of meat offered to idols (1 Cor 8:4 ff, emphasis mine):

⁴ So then, about eating food sacrificed to idols: We know that “An idol is nothing at all in the world” and that “There is no God but one.” ⁵ For even if there are so-called gods, whether in heaven or on earth (as indeed there are many “gods” and many “lords”), ⁶ yet for us there is but one God, the Father, from whom all things came and for whom we live; and there is but one Lord, Jesus Christ, through whom all things came and through whom we live.

⁷ But not everyone possesses this knowledge. Some people are still so accustomed to idols that when they eat sacrificial food they think of it as having been sacrificed to a god, and since their conscience is weak, it is defiled. ⁸ But food does not bring us near to God; we are no worse if we do not eat, and no better if we do.

And again, Paul addresses the issue in the letter to the Colossians Col 2:16-17 and 20-23:

¹⁶ Therefore do not let anyone judge you by what you eat or drink, or with regard to a religious festival, a New Moon celebration or a Sabbath day. ¹⁷ These are a shadow of the things that were to come; the reality, however, is found in Christ.

...

²⁰ Since you died with Christ to the elemental spiritual forces of this world, why, as though you still belonged to the world, do you submit to its rules: ²¹ “Do not handle! Do not taste! Do not touch!”? ²² These rules, which have to do with things that are all destined to perish with use, are based on merely human commands and teachings. ²³ Such regulations indeed have an appearance of wisdom, with their self-imposed worship, their false humility and their harsh treatment of the body, but they lack any value in restraining sensual indulgence.

I don't know that it get's much clearer than this in scripture.

The conclusion I've reached, and this is very much my opinion, is that the need for rules one way or the other reflects spiritual immaturity. The inability to make the leap, without a specific command in the affirmative, from the Old Testament use of instruments in worship to their application under the new covenant in Christ seems, to me, particularly illustrative of such insufficiency.

So my TL;DR answer to your question is, "yes, if something is not specifically permitted we may be able to do it; likewise if something is not specifically prohibited we may be able to do it".

(For more on the use of instruments in worship, see also this answer)

My school has tried to teach that since we are not instructed to use instruments, we CANNOT use instruments.

The same holds for automobiles, drugs and medicine, tractors, refrigerators, computers and so on. Ironically, nobody was told to live in America, as far as I know.

Hence, by the same logic, you mustn't use all that and much much more. In effect, it turns out one must live the life of a nomad, who wanders around with his cattle like Abraham did a few thousands of years.

If you feel like this is sensible, then please do it. But the fact that you used a service you were not told to use (i.e. stack exchange) makes it clear that you've already decided the matter for yourself.