Martin Worthington of SOAS has put online an archive of recordings of “modern Assyriologists reading ancient Babylonian and Assyrian poetry and literature aloud in the original language.” The About page explains:
It is intended to serve several purposes, some for Assyriologists, and some for the wider public. First, it aims to foster interest among students of Babylonia and Assyria in how these civilisations’ works of verbal art were read aloud in the past, and how they should be read aloud today.
Second, it provides a forum in which scholars who have theories about Babylonian and Assyrian pronunciation, metre, etc. can present a concrete example of how their theories sound in practice. [...]
Third, as a record of the ways in which contemporary scholars read Babylonian and Assyrian, it will some day serve a historical function. Many great Assyriologists, including some who had influential theories of Babylonian metre and phonology, passed into history without leaving a single recording of how they read Babylonian and Assyrian. This archive will provide at least some record of how scholars read Babylonian and Assyrian in the twenty-first century.
Finally, but not least, the questions which students of ancient languages most frequently hear from laymen are: “How did they sound? And how do you know?”. This website is meant to serve as an introduction to these issues, providing the public with some idea of how modern Assyriologists think Babylonian and Assyrian were pronounced.
The recordings are here; start by clicking on the open-book symbol at the right of the recording you’re interested in, then open the “listen to the recording” link in a new tab or window, and you can hear the recording while following along in the transcription and translation. It’s a lot of fun, and the ones I’ve listened to have sounded quite convincing (in the sense that they sound like an actual language being spoken rather than the kind of stilted, exaggerated reading you often get from professors trying to read Ancient Greek or Old English aloud—of course, it is extremely unlikely that any of the readers actually sound like an ancient Babylonian). Via Languages Direct; they remark “you are struck, more than anything, by a feeling of familiarity through the similarity in sound of Babylonian to its modern offspring such as Arabic,” which doubtless has to do with the fact that the readers are familiar with Arabic but have never heard any speakers of Ancient Babylonian. Thanks for the link, Sashura!