“Nightswimming” by R.E.M.
I have to agree with these two suggestions.
For Ben Folds check out *Ben Folds Live * which is just him and the piano. Before I saw him live last year I had only heard “Brick” and I had a great time at his concert. I even liked it enough to buy the live album. His dialog is funny and he is a great pianist and song writer.
Tori Amos is a polarizing individual, people usually really, really, love her or hate her with all of the fury of a scorned woman. I happen to like her. For good songs with just her and the piano check out Little Earthquakes, Under the Pink, and Boys for Pele. Her later music has the full band on most of the songs.
I would also check out a few Peter Gabriel songs. Check out “The Drop” from Up, “Here Comes the Flood” from Peter Gabriel 1.
Mike
Q: Who’s alternative and sings over a piano?
A: Sinead O’connor on a bender!
You might want to try some stuff by Color Theory.
I recommend “The Perfect Song”, “Now I know”, and the live version of “Heart Like a Doll”.
Ah, Tori’s wonderful, just a bit wacky is all.
Peter Gabriel is my personal savior. 
“The Drop” is just piano and voice, but “Here Comes the Flood” has a full band. There are probably several versions floating around on bootleg that are just him and piano. I know he did a keyboard/voice version on a Kate Bush special many decades ago (seems like).
Speaking of, Kate Bush has a bunch of wonderful piano/voice songs too. One gorgeous song that’s just piano/voice is called “Warm and Soothing.” It’s a b-side. I could help with an mp3 if need be.
Other Kate Bush choices,
“In The Warm Room”
“Oh England, My Lionheart”
“And Dream of Sheep” (has a tiny bit of guitar at the end, and sea sound effects)
“Feel It”
Every single “Cathy Demo”!
Of course, SmackFu might not be interested in the Peter Gabriel/Kate Bush/Happy Rhodes/Tori Amos style of “alternative.” It’s hard to know.
How about something from Ben Kweller’s Sha Sha. I know the title song is just Ben and the piano. One of the more well-known songs on the album, “Wasted and Ready” has other instruments.
Thanks for all the suggestions! I’ll have to check them out on Napster (legally of course).
The Ben Folds Live album is great. Especially the part of Army where he has the audience do the horns.
I agree with you on the Army bit, SmackFu. I heard that CD one day when I was studying in a cafe. The horn/audience part amazed me. I became a BF fan instantly, and I was really excited this past April when I got to be a part of that “horn” section at his concert. It was also really cool when we did the “ahhh” part on Not the Same. Amazing how an auditorium full of regular people can actually sound pretty good when they all sing together. 
If you want Ben Folds without the Five, I highly recommend his EPs “Sunny 16” and “Speed Graphic.” Very, very good.
Smackfu: You should try Warren Zevon if you haven’t already listened to him. He was a virtuoso piano player, and studied music under Igor Stravinsky. Piano is featured in most of his music, and his live stuff consists of mainly him and a piano or occasionally a guitar.
Thats a great song!
Check out The Clash’s cover of “Every Little Bit Hurts” from the Clash On Broadway box set. I love that song, and I’m not a guy who likes piano based sappy love type songs.
I’d throw another one in for Rufus Wainwright, and am surprised only one other person mentioned him. The most “piano only” songs are probably on his eponymous debut album, the two that followed have been more and more heavily produced.
As far as an individual track, “Complainte de la Butte” off of the Moulin Rouge soundtrack is (if I remember it correctly) entirely him and his piano, and entirely breathtaking.
Off the debut album, I’d give a go with “Baby,” “Damned Ladies,” and most strongly “Foolish Love”
From Poses, “In A Graveyard,” “Tower of Learning” (though further instruments kick in heavy for the second half on that one), and definitely the title track “Poses” above all else.
And from the most recent, Want One, “Dinner at Eight” and “Pretty Things” are both good bets.
On other fronts, I’d definitely give a listen to some of the more obscure stuff Elton John has put out over the years, some of it’s pretty crazy.
I don’t know if it’s really your type of music, but since you said “alternative” you can try Something Corporate. I guess they can be described as “punk with a piano” but I use the term punk in a loose sense seeing as how it’s been bastardized in today’s music industry anyway. The live show is amazing though, the guy really bangs on the keys.
There are a couple of good Radiohead songs that fit the bill. And I think Thom Yorke is a very big Nick Drake, who I haven’t heard but (I think) might also qualify, based on what I’ve read.