Not much is known in a specific fashion about ancient Greek music; however, some tantalizingly interesting material is available on line at a site on Ancient Greek Music, copyright by Stefan Hagel. The first musical passage which is available here is the original music for Euripides, Orestes, 338-344, and there are also a number of later Greek musical fragments.
More speculative is the nature of Homeric recitation. An interesting reconstruction of what a Homeric passage sounded like, as originally performed, is available from Georg Danek and Stefan Hagel, Homeric Singing - An Approach to the Original Performance. This includes a recitation, in an attempt at the original Greek pronunciation and musical accompaniment, of Demodokos' song about Ares and Aphrodite: Odyssey 8, 267-366 (for an English translation of this, see H & P, pp. 162-164; H & P give the line nos. as 1-109).