jsgoddess, have you heard of a band from Columbus called Earwig? I like them pretty well.
Also, check out The Black Keys. They are awesome.
And Be Your Own Pet, Mew, and of course Pavement
Where is Argent Towers when you need him? He can name every indie band in the country.
Both of them? Wow. That really is identical!
Try some Todd Snider. He is a filthy hippie but has some pretty good tunes.
I think the OP means indie as “eclectic non top 40 shit most people haven’t heard of”. “Indie” seems like it is pretty much used these days the same way “alternative” was used in the 90s and “college rock” was before that. Technically **The White Stripes, The Strokes, The Shins, The Killers, The New Pornographers, The Bravery, The Arctic Monkeys **and **The Franz Ferdinand **are part of the “indie” genre, but they are probably a lot more mainstream than what the OP is looking for. **The Shins **stopped being “indie” when they began to appear on the soundtrack to every movie or TV show Zach Braff was ever in.
Windwalker, How many Irish people named **Rodrigo y Gabriela **do you know? They’re Mexican. Although I believe they did spend a lot of time touring in Ireland. They do a really good cover of Metalica’s Orion.
I saw **Rodrigo y Gabriela, Cat Power, Rouge Wave **and **Tray Anastasio **(the dude from Phish) play at the All Points West festival (kind of an indie/jam band festival they threw in Jersey City, NJ over the summer). They were all pretty good, although I really hadn’t heard of them before (I was there more to see Jack Johnson since Radiohead was sold out).
If you like Imogen Heap, you might also like Sia (also a popular TV show soundtrack artist,** Frou Frou, Sigu Rós, Massive Attack, Faithless **(basically Dido + a black hip hop dude), and Portishead, Thievery Corporation, Zero 7 and, of course, Bjork. They all have kind of a similar haunting, electronic, trip-hop, mellow female vocal sound. Again, probably not true “indie”, but whatever.
Maybe Badly Drawn Boy?
Ah my bad. My Irish friend hooked me on to them and he said they had immigrated. Wouldn’t really surprise me much anymore, though, what with Swedish singers named José Gonzales and what not. 
I really can’t recommend Fleet Foxes enough. I think both the album and the EP are great all the way through. They have maybe three weak tracks between them.
How 'bout real, real indy? 
My friends (acquaintances, really) Coyote Grace are coming to your…uh…state… (I don’t know if it’s your *town *(Cleveland)), tomorrow! I finally got to catch their show last night and they’re really sweet and wonderful. They have some stuff to listen to on their website to see if they’re up your alley.
Just a girl with an upright bass (and occasionally ukelele) and a boy-who-used-to-be-a-girl with a guitar. Joe’s got a real lyrical gift, and Ingrid’s a sassy brassy occasionally-into-twangy Southeastern Ohio gal. I don’t think I’ve ever heard a cover of “I’m on Fire” that was quite so hot!
Ahmond
Sonia / Disappear Fear
Dan Bern
Moxy Fruvous / Jian
Gonna jump in with a few more recs:
Polyphonic Spree - Chances are if you like The Flaming Lips, you’ll probably find these guys appealing, as well. A lot of dreamy wistfulness with some more up-tempo stuff thrown in as well. They use a variety of instruments and effects, but they don’t ever lose that human feeling.
Johnny Flynn - A Larum is a beautiful debut folk album from this (former?) London actor. Good vocal harmonies.
Jim White - Soft, guitar-driven, and hypnotic. For some songs, he kinda reminds me of Leonard Cohen. As for albums, try Drill A Hole In That Substrate And Tell Me What You See and Wrong Eyed Jesus.
M. Ward - Singer-songwriter from Portland. Folky with a sometimes blues-y feel, his stuff is usually pretty low-fi, but he has a really good ear for when he does put in effects. Like Jim White, he has a soft, hypnotic singing voice. I’ve listened to Post-War and Transfiguration of Vincent, which I both like.
I really want to like Fleet Foxes. I’ve listened to their album several times. I’ve seen them live twice. You can download an mp3 of Wilco and Fleet Foxes singing I Shall Be Released here if you promise to vote. They seem to be really nice guys, really humble, and Jeff Tweedy appears to love them, so I feel like I should love them, too.
And yet…they mostly leave me cold. I don’t know.
Mellow and folksy? The Acorn.
And thanks to looking up their page, I found out they FINALLY have an upcoming show in St. Louis! Thanks!
Oh, and I also second the upthread nomination for The Weakerthans. Really clever lyrics there.
Don’t feel bad about this, I had the exact same reaction to them. There is no doubt that the music is wonderful, but I still feel incredibly “meh” about it. It’s good but, idk, boring.
To the OP, I should add Devendra Banhart to the list. His music has been called “freak folk”, a fitting label that Banhart hates. His album Smoky Rolls Down Thunder Canyon is a wonderful journey through many varied musical styles anchored by his crooner voice.
Hope you enjoy.
Their offical website says they are Dublin-based.
Several posters above have mentioned him and I’d recommend any of Josh Ritter’s albums. You can purchase them on emusic and elsewhere online. You can sample his music here.