Diablo – transfer characters from one region to another in Diablo 3

Yes it is possible. The client is not account specific.

You can use MPQ Explorer.

This lets you open Blizzard's MPQ file format, which is where they store all their game assets including music. The application has the ability to extract sound files from the MPQ files.

http://www.ehow.com/how_4705307_extract-music-world-warcraft.html provides an example on how to do it on World of Warcraft. The only issue is that I don't think anyone has bothered finding which MPQ contains Diablo III's music yet, but there is only a limited number of options.