I believe it is still quite a hard thing to be truly certain of but there are some indications around. Harmonia Mundi produced an album a few years ago that included some ancient Greek music [1].
There is an audiobook called How to Listen to and Understand Great Music. The author starts way way back with the oldest music we have and then little by little moves forward to more modern times showing how music grew little by little over the millennia. The audiobooks includes audio excerpts of the music. I found it fascinating up until the classical music time when it started bogging down and going a lot slower.
We have the notation for a small amount of Hellenistic and Roman era music, and I think a tiny little bit of Euripides. We'll never know what Sappho sounded like, though.
