Or not, as it turns out. The image quality is unspeakably bad, with really pixellated video. It's doubly disappointing because when I have my laptop connected up to my TV to access the same streaming service, the picture is generally pretty reasonable.
The WiiPlayer (as I have decided to dub it) is also a total pain in the arse to use. You have to access the videos using the Wii's Internet browser, which isn't that great itself, rather than simply selecting an iPlayer channel from the main menu as you might expect. The constant scrolling around the iPlayer Web page you have to do using the Wiimote isn't that user-friendly either.
The final nail in the coffin is that the video doesn't fill my TV screen. Unlike the PC version of the player, there's no 'full screen' button. You can zoom into the video, but you still get irritating bars on either side of the screen.
Now to be fair, this is all still in beta, so hopefully all these issues will be addressed over time, but one thing will be difficult to overcome is that the Wii is the worst of all the most recent consoles when it comes to the quality of the video connection.
The Xbox 360 Elite and the PS3 both have HDMI ports -- the standard connection with a Wii is composite video, which might explain some of the fuzziness on the video I was watching.
I've now invested in a cheap component cable from Amazon, which will hopefully improve things, but I'm not expecting it to be Earth-shattering. I'll keep you updated if I notice anything significant when it arrives, and whether the BBC's update to its WiiPlayer make it any more useful.
Update: The good news is that the component cable arrived, but unfortunately the picture quality is just as crap as before. Here's hoping it will get better soon.