Dood's Music Streamer and TinyShark are two apps that let you stream and download music from Grooveshark for free. You don't even need a Grooveshark account!
Dood's Music Streamer used to be the better app, but a new layout and badly placed ads were not an improvement at all.
TinyShark is a pretty good Grooveshark client too, even though its user interface is rather clunky. It doesn't need Adobe Flash anymore, and it can now scrobble your tracks to last.fm.
March 1, 2012 update: Dood's Music Streamer is updated and works again.
September 4, 2012 update: Last time I checked TinyShark was no longer available in the Google Play Store. But you can download the APK elsewhere. The app still works for now, until Grooveshark changes their setup again.
October 19, 2012 update: Dood's Music Streamer got booted from the Google Play Store, but you can download it straight from the developers website.
• Dood's Music Streamer straight from its maker
• TinyShark (Exigo Software)