Oh, I have an answer! But it makes me sound nicely creepy. :)

I've actually got an entire book related to this called "Greek and Roman Necromancy", by Daniel Ogden. The term "Necromancy" in this case means telling the future through communication with the dead, which would be accomplished through things like looking at entrails, sleeping on tombs, etc.

All that would indicate a belief in some form of predestination. But there's also another book, "Curse a Tablets and Binding Spells from the Ancient World". Those were usually tablets called defixiones inscribed and thrown into wells or placed in cemeteries, which might indicate some flexibility in predestination- an ability to bargain with fate.

So in answer, people's beliefs were probably complex and fell accross a spectrum, with some feeling strongly about the immutability of the future and others not so much.