The concept of "atheism" as we conceive of it is a very recent idea that resulted from the 18th Century's growing concept of a division between secular and religious realms of human experience. This itself was something that had been gradually bubbling up since the Renaissance, but it wasn't something that a person living in the ancient world would've understood. For us the concept of religion is the direct opposite of the secular, and there are lots of sharp distinctions that separate the two of them. Not so in antiquity. For a person living in that kind of society the way we think of religion in the modern west is entirely alien. The concept of doctrine being binding, for example, wouldn't make any sense to an ancient observer, who would probably find the insistence of religious authorities on a single true statement about the qualities of a divine power as being ridiculous. The division between secular and religious would've also perplexed an ancient observer, since religious rituals and secular rituals are one and the same, with no distinction (this is something that we've forgotten in the west, but which is still understood and present in many other religious traditions--such as those of Japan and China). And that ritual aspect of religion is important, since the concept of religious observance by faith alone is completely bizarre in antiquity and doesn't exist until Christianity starts to really take hold. What's important in ancient religions--and still in some, such as Shintoism--is the observance of the ritual. It really doesn't matter what you believe, so long as you perform the ritual. In reality it's a bit more complicated than that, but it's also the subject of an awful lot of very wordy books that I don't have the space or time to summarize here.

So the simple act of not believing in gods or in the traditions that were connected to them was not really a big deal. The Epicureans, for example, held that there were no gods that could be held higher than humans (although many Epicureans very much accepted the concept of gods and immortal beings--again with the lack of doctrine in religion, even though it could exist in philosophy), and they never really got into any serious problems. The issue was not what you believed, but what you did. Failure to carry out the necessary rituals of the state and so forth was a serious matter, since it could bring pollution on the population. The crime of the Jews was not carrying out the rituals due to the Emperor, not their belief in a single god. Now, since these rituals were a very fundamental part of the way societies worked it was very hard for someone to exist and not participate in them.

But what about the charge of "atheism" that was leveled at Socrates. For some reason people like to pick up on this one and ignore the rest of the charge. Socrates wasn't charged with atheism as we understand it--he was charged with disrespecting and refusing to accept the gods of the city. That's exactly the same thing as refusing to accept the city itself, since the state gods are the city and its people rolled up into one. In short, Socrates was being charged with treason, not atheism as we understand it. In any case, as Xenophon makes clear, these charges were ridiculous.

I don't think I've really answered your question very well, mostly because without a good grounding in just how people of antiquity really understood religion it's not possible to describe idiosyncrasies in ritual observance like this, but I've given it a shot. I strongly suggest you take a look at the works of Nilsson, Burkert, and more recently Zaidman's work on the ritual observance due to the chief gods of the city

I cannot speak to the Greek point of view on this at all - /u/XenophonTheAthenian did an excellent rundown, so I'll leave you to read that.

With Romans, it would depend on how you approached your atheism. To draw an analogy to the modern day - I know some atheists who are, like, actively atheist but still celebrate Christmas as a cultural, rather than religious, holiday. Similarly, I know atheists who are culturally Jewish and attend family seders at Passover, etc. If you were this kind of atheist - happy to participate in the religious events that were a part of the cultural context of wherever you lived - nobody would have said anything to you. Romans as a whole were generally pretty comfortable with differing belief systems, because their empire was so vast. They also had the concept of the household gods, the lares - gods who were specific to a family. Devotions to the lares were done privately. If you were happy to be firmly atheist but not be the atheist version of a bible-thumper, instead adopting the Roman attitude of "Eh, it might be weird, but it's his thing, so I'm gonna let him do that," you were gonna be treated in the same way.

Where this came into conflict was the emperor cult, and this is where the - I hesitate to use the word "myth" so let's call it an idea - of Christian persecution came from.

The Emperor cult was a method of showing devotion to the Roman state, and it's the one cult that was consistent across the Empire. A few times a year, on specific holidays, citizens would perform devotions to the deified emperors - so, the emperors who had died and been declared gods by their heirs. This was mandatory, except in the cases of religions with special permission not to participate (i.e. the Jews). It generally involved an offering of incense.

Now, the kind of atheist who celebrates Christmas or goes to a seder to keep his parents happy would be fine. Go to the temple with your family, light some incense, leave. Whatever.

This, however, is where the Christians ran into trouble, because they flat-out refused to participate. Because they were viewed as a cult, rather than a religious sect, they did not get an exemption, so they had a choice - suck it up and light the incense, or face prosecution.

We do have a record of one of these trials, by the way, and it's hilarious. The judges are so completely done with the Christian who's on trial, and are basically trying everything they can to get him to see that he's being silly. He's demanding that they execute him, and they're like, "Okay, but... just a pinch of incense? You're a Roman! It's what we do! Don't you care about the emperor's health? Here, I will give you the incense, all you have to do is light it. We can all go home."

So if you're the kind of atheist who would be like, "This is dumb, and you're dumb, and I'm gonna roll my eyes the entire way through this to let you know how dumb I think you are, but I'll still do it because it's a cultural thing and I'm not trying to die over your fake religion" then you'd be ok. If you're gonna demand to be executed for your principles, the Romans would oblige you. They'd try to talk you out of it, because you're worth more to them as a participatory member of society than as a corpse, but the laws are the laws.

Since you specifically asked about BC, and the phenomenon of the emperor cult didn't really start until after Augustus, I will note that before the advent of the emperor cult the Romans were generally pretty live-and-let-live when it came to religion or lack thereof. As long as you weren't doing anything to actively piss off the local gods, you were fine. So again, it depends on what sort of atheist you were.

For a specific example, you might want to look towards the Apology by Plato. It is an account of Socrates' trial before his death, where he is accused of corrupting the youth, as well as "not believing in the gods in whom the city believes".

While there are numerous reasons why Socrates was put to trial and death, this is one of the charges brought against him, which at least shows it was considered a crime of some sort.

I believe Xenophon's account of the same event also mentions this as one of Socrates' charges.

Actually there is one historical figure who fits the description of what we would call an atheist. However, there are few sources on his life and most of these come from long after his death. Diagoras of Melos, who was a student of Democritus, was famous for claiming that there were no gods. The Athenians formally charged him with revealing the Eleusinian and other mysteries to the uninitiated. It was also claimed that he chopped up a wooden statue of Hercules to make firewood and used it to cook turnips. As for punishment, the Athenians offered one silver talent for killing him, and two talents for catching him alive. Diagoras escaped Athens, however, and is thought to have died a natural death in Corinth.

I'm not a historian, but I do know something about this, as the interface of Christianity and Roman culture is an interest for me.

"Atheism" in that era wasn't considered a religious belief in the absence of deity, but rather a failure to adequately honor the local deities. For two hundred years or so, Christians were considered atheists. Judaism was marginal; its antiquity led the Romans to give it a pass for a while, but it was always a little suspicious. (Eventually the Romans got sick of the Jews and tore down their temple. The Arch of Titus in Rome depicts this event, and the spoils were used to build the Colosseum.)

Religion in the ancient world, especially polytheistic religion, was generally oriented toward local or tribal deities. Each locality (city, region, country) had a set of gods that they believed showed special favor toward that region. In fact, the region's prosperity overall depended on the deities' continued good will. In Rome, this was Jupiter, while in Ephesus, it was Artemis. Furthermore, smaller places (e.g. forests, groves, etc) had their own gods and spirits, which could be equally happy or annoyed. Even after the major polytheistic religions died out with the spread of Christianity in Europe, belief in these sort non-god, non-animal nature spirits continued until the modern era.

Individually, some Romans and Greeks had strong devotion to a particular deity, usually in response to favors granted in the past. For instance, a man who believed that Diana had healed his disease or gave him a strong harvest might devote himself and some of his resources to worship of Diana. This doesn't mean that he didn't acknowledge the existence of other gods, though, which is important to avoid a charge of atheism.

With that background, you can probably guess how a people would respond to an atheist. Since an avowed atheist would probably annoy the city's protector gods, responses were generally hostile. Participation in religious rites was generally considered a civic duty, even if your primary religious devotion was to another god. Those leading the civic rites were generally elected by the population or appointed by its leaders, and were considered to hold political offices. People who didn't participate could be barred from political practice, driven out of town, or outright killed.

As others have noted, this changed dramatically depending on the region and the time period in question. "Greek / Roman" spans an area the size of the United States and almost a millennium.

(I'm getting most of my information from Among the Gentiles: Greco-Roman Religion and Christianity by Luke Timothy Johnson.)

This may be controversial. I don't know. Socrates seems to promote the gods as being representative of various concepts, pursuits or facets of nature. He seems to talk about them as representing different values or ideals, not being actual beings. I would not be uncomfortable suggesting he was killed for being a closet atheist.

I think his message caught on, and it became socially acceptable to think of the pantheon as literal beings or as representative of values/facets of nature after that.

Another story that I can think of that should shed light on this is the martyrdom of Polycarp. He was put on trial, and told that he could go free if he said "away with the atheists". (Christians were called atheists because they didn't have statues or any physical representations of their God). Polycarp gestured toward the people who were trying him and said "away with the atheists". Theologocially, Polycarp thought of the Romans as atheists because they didn't have a creator-god. All their gods were gods of facets of nature, in a sense.

He was burned because he would not call Christians atheists while on trial.