As Tunein refused to release an app for the Playbook you can try Norbex. It's a free radio program that will allow you to listen to radio stations within your own country. Which, oddly enough for Canadians, means you can listen to radio stations in Canada and the United States.
If you want to go international, you will need the premium version which will cost you some amount (I dont remember). I have never been able to figure out how to pay for the premium edition through App World on the Playbook - I get an error message each time.
Anyways, if you download Norbex, you can search for CBC and listen to their radio streams using this app. The app is actually quite nice; I just wish that I could get international stations on it.
I listen to CBC Music every day, all day. I simply use the browser & have saved the link to my Home Screen. This works far better than the CBC Music app that I have on my iPhone 4s (the actual webpage has more options and grants access to more music).
And if you open another app, it continues to play? I would prefer to use the TuneIn Radio app that I use on my iPhone on my PlayBook, but it's not available in the appworld.