Nowadays, most classicists seem to teach a reconstructed Ancient Greek pronunciation, imitating how an Athenian would have spoken in the fifth century BCE. On top of that, there's solid evidence for how other dialects sounded, and for sound changes after the fifth century.
But what about before the fifth century? How far back can pronunciation be reconstructed with any certainty? Is it possible to reconstruct a proper Homeric pronunciation for the Iliad, for example? And if not, how close can we get?