Thanks to the efforts of scholars at Cambridge, you can now hear what it might have sounded like. (Obviously, there's no way to know with any certainty how Akkadian sounded, but certain clues, comparisons, and good guesses can give us an idea.) Here is a link to an MP3 file on Slaw's server; you can also access the recording on the Cambridge site.
The excerpt is taken from the Epilogue to the Code, a paean of praise to the ruler (and put in Hammurabi's mouth) that might indeed be justifiable.
Except for those among you who had been waiting for just this stimulus to learn Akkadian, this has a curiosity value. Still, it's interesting to hear law talking across the millennia. And speaking of hearing, I hear all sorts of other languages in this recording … a little Mandarin, some Norwegian, and a fair bit of Klingon.