Annwn is a recently disbanded Celtic rock band in the SF bay area. Great group, lots of good songs, but I’ve never heard them on the radio, not even the local stations. They are on a independent label called Elfhill Music. You can listen to some downloads from their last album, Anarchy & Rapture on Amazon. Pity really. I like their next-to-last album, Come Away to the Hills, much better.
I heard Modest Mouse’s “Float On” on the radio once around here (DC area).
Possibly in WPLJ in New York. 
It depends on how you define “radio.” Most of my favorite groups could be found on college radio and FM radio in the 70s. But you don’t hear them nowadays.
I’d say the biggest group not to get any airplay of all currently is the Band. They were one of the top ten groups in the early 70s, but never had a hit single, so you don’t hear them anywhere any more. And it’s not like their music is dated or anything.
If you counted local groups, I’d include the McKrells (Celtic bluegrass 
).
And some other great (though not commercial) musicians:
Otis Taylor
Deborah Coleman
Siegel Schwall Band
Bonzo Dog Band
Tom Lehrer
Great band, but since a lot of their other stuff sounds like someone throwing a handful of magnetic poetry at a wall and then singing it with tourettes like invections of profanity, you won’t hear much of it on the radio.
That said, go out and buy McLusky Do Dallas.
Supergrass
My Morning Jacket (incidentally, the world’s best live band)
Built To Spill
Neutral Milk Hotel
I should go check my iPod, there’s tons more…
Meanwhile, here in Chicago you can’t get away from it. I swear that song and Franz Ferdinand’s “Take Me Out” is on a neverending loop on WXRT (93.1).
Never Heard on the Radio
King Crimson
Spock’s Beard
Porcupine Tree - my favorite band
Braindance
Marillion (the older stuff)
Maybe once or haven’t heard them in years on the radio
Dream Theater - haven’t had any play since 1993
Ben Folds - I’ve heard Brick years ago, nothing recently
Tori Amos - Used to hear her all of the time in the early-mid 1990s, nothing now
And does it count if I saw the video for that at Target once?
Long long ago (ok, five years) we had the most blissfully wonderful indie radio station here that played all kinds of stuff you would never hear on the radio. I’m betting that some of your CD collection was in their rotation, because metalheads all over Indianapolis sat down and cried the day they got bought out. Ah, Rock 107, we barely knew yee.
Have you ever listened to the internet station Tundra Trash? They play all the cool metal and other heavy stuff. Check it out.
Being from NB, Canada, I heard lots of Sloan and Moist/Dave Usher on the radio.
I haven’t heard that much of either on the radio lately, though since moving to Seattle, I have few complaints, as I hear a much wider variety and receive far more radio stations than I did back in my small town. And there’s a cool little college station that plays dance/techno music, so I get to hear one of my favourites, Assemblage 23, often, as Tom Shears seems to be a favourite of theirs, as well.
A few bands I adore but never hear on the radio, ever, are: Air, Stereolab, Postal Service, Aphex Twin, and Her Space Holiday. I’m sure there are many more I’m just not thinking of, these are just off the top of my head.
Never heard of that one, but I’ll check it out when I get a chance. There’s no shortage of good internet metal stations, but that’s not what this thread is about.
Another vote for Tori Amos and I add Kate Bush.
This confirms it. Radio stations are weird. ** Hunter Hawk** hears half the bands I don’t. Meanwhile, on the stations I listen to Air, Postal Service, Supergrass, Rammstein, and My Morning Jacket all get airplay. And they play the hell out of Rage Against The Machine, Modest Mouse, Marilyn Manson. In fact on the top 5 at 5 countdown, both of the last two have been on it within the past month, and are heavy rotation as well.
Neutral Milk Hotel
Ted Leo and the Pharmacists
CKY (though 96 Quite Bitter Beings was in everrrrything)
Reggie and the Full Effect
Appleseed Cast
The Good Life
Rilo Kiley
Dntel
Spare me your EMO labels, please.
You’re in Seattle? KEXP is 90.3 and 91.7. They play all of those bands.
(No, I’m not affiliated with the station; I just like it a lot.)
Tori Amos was all over the radio when I was in high school in Columbus, OH.
For a band of their importance, it is very rare to hear Peter Gabriel era Genesis on the radio. They don’t fit the “classic rock” strict format very well. Plus, most of their important songs are around 7 minutes long.
In the past ten years, I think I’ve heard I Know What I like and The Lamb
Lies Down on Broadway on the radio. How I hate corporate radio.
One of my best friends is a fan and another poster on the board, I can’t remember who at the moment, gave me a long list of bands to listen to once s/he found out I liked them so they seem to be a bit of an underground hit.
Have you seen the movie Garden State? It’s where I discovered them… wonderful movie and soundtrack.
I have Goldfrapp saved in a word.doc on bands and books I should try. Thanks.
I’ve never heard Self on the radio, despite the fact that they had songs on the soundtracks for Shrek and Dead Man on Campus. It’s a shame too, because I think this is a band that could be really popular if they were given airplay in the right markets.
Also, I’ve never personally heard Sleater-Kinney on the radio, but I know they do get airplay in other places.
Lean Andreone
Chantal Kreviazuk
The Hail Marys
PJ Harvey
Bjork
Another vote for Motorhead
scarling.
Hole
Courtney Love [solo]
I’ve also never heard The Ramones or Joey Ramone [solo] on the radio. I know that’s just my personal experience, though… I’m sure they’re played lots elsewhere.