I'd suggest on your old computer do a computer search .. search .mp3 and any other music file formats such as .wma, .mp4, .m4a, .ogg, .wav, etc.
Then transfer them to the new computer.
Also see
If you updated LimeWire & cannot find your Downloads. Oh .. and check your normal music folder such as C:\Music or C:\Documents and Settings\User-name\My Documents\My Music