Before 1980, was there ever an attempt to evangelise for Orthodox Christianity?

Did the Byzantines or Russians try to convert their conquered subjects to Orthodoxy ?

During the Russian conquest of Siberia

In Kamchatka, the Russians crushed the Itelmens uprisings against their rule in 1706, 1731, and 1741. [...] The Russian Cossacks also faced fierce resistance and were forced to give up when trying unsuccessfully to wipe out the Chukchi through genocide in 1729, 1730–1, and 1744–7. After the Russian defeat in 1729 at Chukchi hands, the Russian commander Major Pavlutskiy was responsible for the Russian war against the Chukchi and the mass slaughters and enslavement of Chukchi women and children in 1730–31, but his cruelty only made the Chukchis fight more fiercely. A genocide of the Chukchis and Koraks was ordered by Empress Elizabeth in 1742 to totally expel them from their native lands and erase their culture through war. The command was that the natives be "totally extirpated" with Pavlutskiy leading again in this war from 1744–47 in which he led to the Cossacks "with the help of Almighty God and to the good fortune of Her Imperial Highness", to slaughter the Chukchi men and enslave their women and children as booty.

The Russians were also launching wars and slaughters against the Koraks in 1744 and 1753–4. After the Russians tried to force the natives to convert to Christianity, the different native peoples like the Koraks, Chukchis, Itelmens, and Yukagirs all united to drive the Russians out of their land in the 1740s. [...] Kamchatka today is European in demographics and culture with only 2.5% of it being native, around 10,000 from a previous number of 150,000, due to the mass slaughters by the Cossacks after its annexation in 1697 of the Itelmen and Koryaks throughout the first decades of Russian rule. The killings by the Russian Cossacks devastated the native peoples of Kamchatka.

According to Western historian James Forsyth, Aleut men in the Aleutians were subjects to the Russians for the first 20 years of Russian rule, as they hunted for the Russians while Aleut women and children were held as captives as a means to maintain this relationship.

The oblastniki in the 19th century among the Russians in Siberia acknowledged that the natives were subjected to immense violent exploitation, and claimed that they would rectify the situation with their proposed regionalist policies.

The Russian colonization of Siberia and conquest of its indigenous peoples has been compared to European colonization of the Americas and its natives, with similar negative impacts on the natives and the appropriation of their land. The Slavic Russians outnumber all of the native peoples in Siberia and its cities except in the Republic of Tuva. [...] The Buriat make up only 29,51% of their own Republic, and the Altai only one-third; the Chukchi, Evenk, Khanti, Mansi, and Nenets are outnumbered by non-natives by 90% of the population.

The Ainu have emphasized that they were the natives of the Kuril islands and that the Japanese and Russians were both invaders. In 2004, the small Ainu community living in Kamchatka Krai wrote a letter to Vladimir Putin [...]. In the letter they blamed both the Japanese, the Tsarist Russians and the Soviets for crimes against the Ainu such as killings and assimilation, and also urged him to recognize the Japanese genocide against the Ainu people, which was turned down by Putin.

Likewise, Ivan the Terrible

was a devoted follower of Christian Orthodoxy, but in his own specific manner. He placed the most emphasis on defending the divine right of the ruler to unlimited power under God. Some scholars explain the sadistic and brutal deeds of Ivan the Terrible with the religious concepts of the 16th century, which included drowning and roasting people alive or torturing victims with boiling or freezing water, corresponding to the torments of Hell. That was consistent with Ivan's view of being God's representative on Earth with a sacred right and duty to punish. He may also have been inspired by the model of Archangel Michael with the idea of divine punishment.

Also,

his anti-Semitism was so fierce that no pragmatic considerations could hold him back. For example, after the capture of Polotsk, all unconverted Jews were drowned, despite their role in the city's economy.

Did the Byzantines or Russians try to convert their conquered subjects to Orthodoxy ?

Britannica informs us that

In Bosnia the Bogomils, equally persecuted by Orthodoxy and Catholicism, had religious as well as material reasons for conversion [to Islam].

More to the point,

Basil the Physician was the Bogomil leader condemned as a heretic by Patriarch Nicholas III of Constantinople and burned at the stake by Byzantine Emperor Alexius I Comnenus.

He first came to the attention of the emperor after imperial officers had tortured a member of the Bogomil sect, named Diblatius, to reveal the identity of their leader. [...] This sect, noted for [...] their detestation of the Orthodox hierarchy, had been rapidly gaining adherents throughout Alexius’ reign, and began to cause alarm among the Byzantine clergy.

with Basil refusing to renounce his opinions, the Patriarch of Constantinople, Nicholas the Grammarian, together with a synod of bishops declared Basil a heretic. Alexius then pronounced a sentence of death on Basil, by being burnt at the stake.

Prior to the [death] sentence being carried out, Alexius attempted on several occasions to have Basil recant, but each time Basil refused to change his mind, stating that angels would descend from heaven to release him from the stake. Finally, Basil was burned as a heretic in the hippodrome of Constantinople.

_{I believe the above to easily qualify as the bloodshed and cultural destruction mentioned by the OP in (t)his question's initial, uncensored draft.}

Much more recently than the other answers: http://www.hchc.edu/missions/articles/articles/orthodox-mission-in-tropical-africa documents Greek Orthodox missionary activity undertaken in sub-Saharan Africa by the Patriarchate of Alexandria; It is pretty much 20th century. The bishoprics of Africa are listed under https://en.wikipedia.org/wiki/Greek_Orthodox_Patriarchate_of_Alexandria_and_all_Africa, and mugshots of the bishops are in http://www.patriarchateofalexandria.com/index.php?module=content&cid=004001 (in Greek).

I don't have a source for this, but what I heard from others is that the methods are different. Orthodox missions consist of some clergy showing up, building a church, and starting to do services, and waiting for people to start showing up.

They will just go ahead and translate the service into the local language, inventing an alphabet if necessary, and start. They don't really do to much "I give you food/medicine, you become Christian?" sort of exchanges. They just do the liturgy and wait.

It is my understanding that in addition to the missions in Alaska, the whole of Siberia was converted this way : build church, translate bible, and wait. (At the dawn of the SU, there was a ton of written languages that got replaced by latinized alphabets, then Russian.)

There is some of this happening in Uganda between 1920 and now. More recently one in Guatemala.

The answer is yes.

Orthodox Christianity has a long history of Missionaries proselytizing and spreading the Christian Gospel to various cultures.

The first example dates to the 800's AD/CE, during the heyday of the Byzantine Empire. There were 2 Greco-Thessalonian Saints named, Cyril and Methodius. One of their major roles or "mission", was to convert the Slavs to Christianity; more specifically, the Slavic peoples living in the neighboring Balkan region, particularly around Bulgaria, whereby many Slavic peoples had settled beginning in the 500's AD/CE.

The Slavs, were originally a polytheistic pagan culture whose geographical origins were in Russia and Poland. However, by the 500's AD/CE-(about a generation after The Fall of the Roman Empire), many Slavic peoples conquered and settled throughout Eastern Europe, which at that time, was largely comprised of three major lands: Illyria, Dacia and Thrace-(though including Thrace as an Eastern European land, is debatable). The Slavs dramatically and irreversibly changed the demographic landscape of Eastern Europe-(as well as having invaded a sizable portion of the Greek mainland, including parts of the Peloponnese in Southern Greece).

After living in the Balkans for 300 years-(including a sizable portion of the Greek mainland), the Greco-Thessalonian Bishops, Cyril and Methodius had assimilated the Balkan Slavs into Byzantine rite Christianity-(what would later be called, "Orthodox Christianity"). As the Balkan Slavs adopted the Byzantine Christian religion and culture passed onto them by Saints Cyril and Methodius, it began to spread and influence the Slavs of Russia whereby the First Russian King-(or one of the earliest Russian Kings) adopted Byzantine rite Christianity at the beginning-(or near the beginning) of the 2nd Millennium AD/CE.

Though the Byzantines also brought their own form of Christianity to Africa, specifically, Ethiopia, centuries before the arrival of Cyril and Methodius. During the early years of the Byzantine Empire-(I believe in the 500's AD/CE if my chronological memory is correct), a Greek Bishop named Frumentis and his Missionary Entourage had introduced Eastern rite Christianity to the Ethiopians. Now, it should be noted that Ethiopia has a long historical connection with the Middle East dating to King Solomon; so the Early Medieval Ethiopian assimilation and conversion to Byzantine rite Christianity was not necessarily that culturally radical or shocking-(and relations between Greeks and Ethiopians predate the Byzantine era, dating back to the days of Homer).

And if my memory is correct, the Byzantines also conducted Christian missionary work in Nubia-(present-day Northern Sudan) centuries before the Sudan was conquered by the Arab Muslims. In other words, "once upon a time", Northern Sudan, was Christian, specifically, Byzantine Christian.

There were perhaps other regions within the Byzantine Empire whereby Missionary activity may have occurred; places, such as Sicily, parts of North Africa, as well as the Middle East. However, Byzantine Christian missionary work within the Balkans and East Africa during the Early Middle Ages is well documented.

Yes, sending out Orthodox missionaries was quite common, at least in the more distant past.

When the Byzantine Empire was still a thing in the Middle Ages, missionaries were commonly sent to the pagans in Eastern Europe and the Middle East. The Slavs in particular were common recipients of missionaries. The missionaries at the time commonly pursued a top-down style of conversion, where the local ruler was converted, who would then convert those beneath him.

The Serbians, Bulgarians, and Wallachians were converted this way, and the Russians were converted as one of the terms of a peace deal. Missionaries were then sent to the Russians to teach them and monitor their conversion.

In fact, an Orthodox missionary from Constantinople, St. Cyril, invented the Cyrillic alphabet while on a mission to what is now the Czech Republic and Slovakia in order to better write Slavic languages. Those regions ultimately ended up becoming Roman Catholic, but Cyrillic flourished in the Slavic Orthodox regions.

Once the Byzantine Empire declined and eventually fell in the late Middle Ages, Russia was the main Orthodox power and was the driver behind further proselytizing. As Russia expanded, missionaries were busy converting the non-Christians in the newly-acquired regions.

There have no doubt been missions to what are currently non-Christian regions, but I have no information on those. My guess is that Russia had missionaries in China in the early 20th century along with all the other European powers. I'm not aware of any general success in conversion in the 20th century. I suspect that Orthodox missionary work greatly declined when most of the Orthodox countries became communist.

Yes, Eastern Orthodox missionaries preached their version of the Gospel to outsiders in a manner not very different to Protestants and Catholics.

Russian missionaries preached Orthodoxy to Alaskan natives in the 1700's (Alaska was by then a territory of the Russian Empire), converting some. Also see here. One famous Russian missionary is St. Innocent of Alaska, who started translating the Bible into Aleut.

The Russian Orthodox Church honors St. Nikolai of Japan, who (in the 1800's) preached Orthodoxy in Japan, which was never part of the Russian or Byzantine empires.

So, the short answer is:

Yes, they attempted to spread their religion to conquered areas within their borders.

Yes, they attempted to spread their religion beyond their borders.