If I read this article correctly, Ajjan composed new/modern pieces based on the ancient hymns in another classic example of contemporary artists being inspired by archaeological objects or ancient history. Another example mentioned in the short article is the performance of a song from the tablet by Syrian Soprano Noma Omran at Daitoku-Ji, a Zen-Buddhist temple in Kyoto, accompanied by the temple's monks and Japanese percussionist Stomu Yamashta. Cool stuff! I wish these recordings and compositions were available.
If you want to hear the actual hymns and not contemporary interpretations, you'll have to go back in time 30 years to a recording entitled Sounds from Silence created by Anne Kilmer, Professor Emeritus at UC Berkeley, and her UCB colleagues Richard Crocker and Robert Brown. I was lucky enough to purchase a copy from Dr. Kilmer herself before she retired. With a little Internet sleuthing though, I was able to find the producer Bit Enki ("house of Enki," the Mesopotamian god of the sweet waters) is still selling copies of the original vinyl and CD on their website at Bella Roma Music.
L to R: Kilmer, Crocker and Brown with replicas of ancient lyres
The songs were produced from replica lyres modeled on ancient examples excavated in southern Mesopotamia. I've used this recording time and again in my classes to give my students a little taste of what these hymns might have sounded like. As Kilmer notes, we have the notes and pitches but we do not have information concerning rhythm or pacing.
Real Player sound clips are available on the Bella Roma Music website but I wanted to include my own little snippet here. This is only a 30 second preview, as this material is still under copyright so I encourage you to purchase the full CD if you are interested!