What program do you use to play music normally? The successful downloads are stored in the "shared" folder on your hard drive, so look in there, drag it onto your music player (the quicktime folder usually has a movie player that will play .mp3's), and you should be good-to-go.

Which version of Limewire are you using? Mine has a "Launch" button on the bottom of the Library pane that works, but nothing in the menus.

Do post back if you find a solution to help others.