This gets brought up a lot, and most of the pieces are mentioned here and there, but I thought I'd put it all down in one place. So:

What you're going to need:

1. some familiarity with the unix shell, and ideally be comfortable with perl and makefiles.
2. a copy of mplayer, compiled for ARM.  I used the one from this thread
3. a copy of get_iplayer (here)
4. a copy of flvstreamer, cross-compiled for ARM (I've got a copy up here, but I'm not warranting to keep it there forever)
5. the perl distribution for chumby, from here

The steps, more or less:

1. install the custom perl distribution for chumby onto your usb key as per the instructions on the wiki: this will put perl in /mnt/usb/usr/bin/perl
2. symlink /psp/usr to /mnt/usb/usr (this is necessary for perl's INC path to work)
3. install mplayer, flvstreamer and get_iplayer into /mnt/storage/local/bin/
4. edit get_iplayer, and change the perl pathname on the first line to be "#!/psp/usr/bin/perl"
5. run get_iplayer:

PATH=/psp/usr/bin:/mnt/storage/local/bin:$PATH USERPROFILE=/mnt/storage  get_iplayer --pid radio:bbc_radio_three --amode=flashaac --stdout --nowrite | mplayer -cache 128 - >/dev/null 2>&1

get_iplayer's startup is, for some reason, really really slow; I haven't yet spent any time trying to profile where it's losing time, as it's kind of a horrible mass of perl.

Obviously running this from the command line is about the least convenient way possible to do it: in theory it should be possible to build a flash wrapper as an External Music Source, but that would require a fair amount of actionscript coding that's currently a bit on the far side of my personal AS learning curve.  I may yet take a stab at it, but anyone who actually knew what they were doing could probably get it done in an afternoon.  I'll send a bottle of nice scotch to anyone who saves me the trouble.

And of course: if you're a chumby user or developer living in the UK, please be sure to let your MP know how grateful you are that the BBC is spending your license fee money for the purpose of making it absurdly hard for you to listen to BBC Radio on any device that hasn't paid a tax to Microsoft or RealNetworks.  A specific demand that blithering moron Ian Hunter be fired with prejudice would not be out of place in my opinion.