I finally got fed up enough with it to look for alternatives, and my choice arrived today: a Sony XDR-S3HD tabletop HD radio receiver. I'd seen it at Fry's and tried it out over at the Sony Style store at the Domain and found it to be really attractive, although I'd not been able to tune HD (hybrid digital, not high definition) signals in the stores -- Fry's didn't hook up the antenna, and the Sony store had it in the back near lots of concrete. However, I thought it had a nice looking wooden case and that speakers sounded great, so I ordered one from the Sony Style Outlet (damaged outer box means $70 discount!).
It showed up this afternoon and I hooked it up about an hour ago It picked up KUT at full signal strength, and I was able to easily tune to the KUT-2 (all talk) and KUT-3 (jazz) substations. I found quite a few other HD stations across the FM dial, and I've got my favorites set as presets now. The sound through the AUX input is also quite nice. The only thing it lacks from the iHome is an iPod dock, but since I don't actually have an iPod, that's no loss for me. I found that it also shows the FM text information for channels that aren't in HD.
If you're curious, here's a useful list of the stations in Austin that broadcast one or more channels in HD.; According to this list, 1300AM is also broadcasting in HD, but the Sony receiver doesn't seem to decode digital signals on that band. No big loss, as I rarely tune to the AM band now that 1600 switched from Air America to Spanish music.