That's what Airplay is for. You browse to the TV station's website on your iPhone, iPod Touch, or iPad in Safari and then Airplay the video to your TV. As long as they aren't requiring Flash. Airplay for video will be available next month.
I'd imagine you would create a folder for each channel on the home screen and bookmarked "web apps" for each show within the folders. Should work out well as I can't imagine watching more than 12 shows per channel regularly.