What you need to do is find a radio-only stream that is playable in Safari, start playing the stream, and then go on to those other apps while listening to your music. As an example, FairerPlatform cites Soma FM, a San Francisco-based, user-supported internet radio station. Their music is available in the pls multimedia playlist format, which the iPad's Safari plays without a hitch. Once you've started the stream, you can do any other work you need to and the stream plays on in the background.
If you're a fan of any of the live shows on the TWIT network, you can listen to those shows in background as well. Just point Safari to live.twit.tv, tap "Live Audio Stream," and then you can listen to their shows while doing real work instead of looking at Leo's face. TWIT AM uses the m3u format, which also loves Mobile Safari. It seems that most of the internet radio streams using pls or m3u work beautifully with Safari.
Of course, this doesn't work with every radio station, internet or otherwise. I tried to see if a local radio station's coverage of the Colorado Rockies would stream -- nope, they use a Flash player. But this is a good trick to know until we're able to run the native streaming players as a background process on the iPad in a few days or months.