I ran backgrounder on my original iPod Touch with no problems whatsoever. Yep, backgrounds and all.
http://www.access-company.com/products/mobile_solutions/netfrontmobile/browser/index.html
The browser on the Sony is capable of loading "full web pages", not a "crappy WAP browser." However, it IS a dumbphone, and I've said this before. My point is a measly little dumbphone can multitask (run internet radio, browse internet, and whatever else at the same time), where an iPhone 3G apparently can't.
It's purely a software limitation. I'm a big fan of desktop OS X, but I think the iOS is a kludged together mess that is getting worse with every revision. It's as if Microsoft took Windows 98 and threw 7's UI on top of it.