This seems to be a Vista/Windows 7 only problem, but I could be wrong on that. It is caused by microsoft updating the Microsoft Synthesizer. Easily fixed though.
1. Install the game from the CD.
2. If you have the retail version download and install the patch (the Sold Out version is already patched):
Mirror 1: http://www.megaupload.com/?d=D74W66CJ
Mirror 2: http://www.4shared.com/file/LwtzSmAg/hitman_sp1aUK.html
Mirror 1: http://www.megaupload.com/?d=8AGCQ85V
Mirror 2: http://www.4shared.com/file/nNfav5Jq/hitman_sp1.html
3. Download this older Microsoft Synthesizer dll:
Mirror 1: http://www.megaupload.com/?d=1RSOO31J
Mirror 2: http://www.4shared.com/file/9aS8DcJI/dmsynth.html
4. Go to the game's install directory (usually C:\Program Files\Eidos Interactive\IO Interactive\Hitman - Codename 47) and put the dll in there.
That should make the sound work again.