Multiple 3-month stays in one Member State with freedom of movement

Can I leave for a period of time (let's say one month) and then have the right to re-enter and stay another month (or three months if needed)?

Basically, the answer is yes. As you may be aware, this is controlled by Directive 2004/38/EC

There's not a whole lot of clarity about this because it's simply not enforced very regularly. In fact, an EU citizen can stay indefinitely in another state subject only to very limited conditions. Among these conditions, the host country can impose a requirement that the EU citizen register. But any penalty for failing to register must be "proportional" to penalties for a country's own citizens failing to register. (This is why the UK does not, strictly speaking, require registration.)

(I assume that the exceptions concerning public safety, public health, and public policy are not in play.)

Accordingly, there's not much reason to go about hunting down people who've failed to register and have been present for more than three months. The benefits simply do not justify the costs. Furthermore, such an enforcement effort would be critically difficult, since the EU does not stamp the passports of EU citizens. It would not be easy to establish the person's whereabouts and the actual duration of their stay. On the other hand, someone who had crossed the border from one country to another, theoretically resetting the three-month period, might have trouble presenting evidence of that fact as well.

Consider someone who is informed by a country's authorities that they've fallen afoul of the registration requirement. Such a person is likely to register, or maybe just leave. Or maybe the authorities would accept the person's assertion of having left and returned, in which case they might enforce a later registration deadline. Regardless, for both parties, going to court would not be worthwhile. So there is no judicial precedent (as far as I'm aware) to help arrive at an answer.

Someone who has been present in an EU country other than their own for less than three months is not subject to the registration requirement, so your question could be rephrased thus:

After staying in an EU country under freedom of movement for three months, what is necessary to reset the three-month period?

The answer probably depends on several factors, and, more importantly, a specific analysis of each case. Consider someone who moves all their things to a home in the country, lives in that home, and then goes away for one month. After the person returns, there is a reasonable argument to be made that the person has exceeded the three-month right of residence. But someone who travels around the country with a backpack for three months could reasonably argue that they are resetting the period with an absence of just a day or two. Furthermore, different EU countries will regard this question differently (remember, the UK doesn't even require registration), so the answer also depends on the country in question.

The foregoing assumes that the person is self sufficient, therefore falling under point 1(b) of Article 7. If that is the case, then the person cannot be expelled from the country nor denied entry. If it is not the case, however, then the person can be expelled from the country. Whether they can subsequently be denied seems to be a matter of dispute. The linked article says that the directive is quite clear that they cannot, yet two governments are trying to create such a possibility (see also Rough sleeping is not an abuse of EU free movement rights).

If someone who was not self-sufficient repeatedly entered a country and became a nuisance, a ban for "abuse of rights" (Article 35) might be imposed. But a second entry, a month after an initial three-month stay, would be very unlikely to rise to that level.