Ensemble de Organographia is a group that specializes in performing music of the ancient world on period instruments. Recently, with some help from archeologists, they have recreated music from ancient Sumer. Here are a couple of videos of what Sumerian music might have sounded like.
They have released an album of this music that also includes music from ancient Egypt and Greece as well. Now perhaps the next time I teach Music Appreciation I should start the historical portion with ancient music, rather than with music from the Middle Ages. I see they also have an earlier album of music from ancient Greece, so I have no excuse now.