What did ancient Babylonian songs sound like? Possibly something close to this. Composer and performer Stef Conner and the Lyre Ensemble have put out an album of songs they believe are close to ancient Babylonian songs, using the original poetry. (Album can be found at that link and will be on iTunes by the end of 2014.) Above, Stef Conner recording, wearing a reconstructed Babylonian gold and lapis lazuli choker necklace. Photo: Stef Conner
The oldest recorded song found in archaeology via a cuneiform tablet is this Hurrian song. It dates from a similar period.
Via here and neighbour/well-known jazz bass musician Torsten Mueller who has played with John Zorn and Arto Lindsay among many others.