No doubt teurastaja is right about the technicalities of why it won't work straight from the browser, but I have found a workaround for anyone interested. It's initially a little bit fiddly, but once you've set it up on your 9300i it seems to work fine.
When you click an audio link on the BBC website it launches the BBC Radio Player - a web page with an embedded real player. But there are also links labelled "listen using stand-alone Real Player" (just below the embedded player). On a desktop PC/Mac, you'll need to make a note of the URLs that these labels point to for any BBC radio live/recorded show you would like to listen to.
So, to take martinh's example of Radio Five Live:
1. On a desktop computer, open up the BBC Radio Player and navigate to Five Live. Hover over "Listen using stand-alone Real Player" with your mouse and make a note of the URL. I get http://www.bbc.co.uk/fivelive/live/surestream_int.ram (might be a slightly different if you are in the UK, due to sports rights).
2. Fire up the built in web browser on your 9300i and enter that URL into it. It will launch Real Player, and, voila, the radio plays. Yay!
3. Now save it and other URLs for future reference.
You can either make them bookmarks in the web browser, or, in Real Player, click stop and then press the menu button/File/Save link as... and give it a name, like BBC Five Live. To find this file later, go to the Desk, then Office, and File manager. The Real Player files are in Communicator/My files. (Must be a quicker way to get to them, but I only bought this phone yesterday...). From there click on the file, real player launches, creates an internet connection and plays the file.
4. Happy listening.