While classics such as Carol (or Canon?) of the Bells are pretty rad, he specifially asked not for them but for alternative.
My favorite Sufjan Christmas song is "Come On! Let's Boogey to the Elf Dance!"
Christmas Cake by Rilo Kiley breaks my heart (so good).
Just Like Christmas and Blue Christmas, both by Low are.... Low doing Christmas songs. Gorgeous to the Nth power.
Christmas Party by the Walkmen is really rad.
Happy Christmas (War is Over) by the Polyphonic Spree gives me chills. Oh Polyphonic Spreeeeee
Christmas Wrapping by The Waitresses is The Best. It's sassy and catchy and rocky with pop.
pas/cal's version of Last Christmas is dreamy.
The Chrismas Song by the Raveonettes sounds like a traditional christmas song, and it's catchy and makes you bop. Whispered vocals are really nice sometimes.
When I Get Home for Christmas by Snow Patrol sounds very much like Snow Patrol. If you like one song of theirs, you probably like them all and I do.
Everything's Gonna Be Cool This Christmas by the Eels is rockin! It's the same as the Snow Patrol thing though, if you like the Eels, you will like this song too. Only it is not so dark!
The twee-est song is Little Donkey by the Boy Least Likely To. It is ADORABLE.
Then of course you have the traditionals sung by indie/alt artists like Feist with Lo: How a Rose E'er Blooming and Clem Snide with Joy to the World and just the really really most awesome christmas song ever (other than the obvious Fairytale of New York) is Johnny Cash's Silent Night.
I really like Christmas.