I was going to suggest the Buckeye Battle Cry by The Ohio State University marching band, but you really can’t dance to it.
But seriously, I’ve been to a few weddings and seen more on TV and in the movies, and my impression was that the first dance song was one that had some personal significance to the bride and groom (e.g., the song playing during an early date, or when they first kissed, etc.).
I tend to be on the same page as you on that one. However, one of my husband’s nephews got married a couple of years ago, and he and his wife had choreographed their first dance, and everyone just sat around staring at each other, disbelieving. It’s hard to do that and not look as though you’re putting on a show… and seeming self-centred as well as ridiculous.
Use Hot Tuna’s “Watch The North Wind Rise”, that’s what we did.
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I was going to suggest the Buckeye Battle Cry by The Ohio State University marching band, but you really can’t dance to it.
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Well, that’s why I said I wanted it to be meaningful and appropriate. There isn’t any one song that leaps out as “our song” or anything, though. We went out dancing a lot when we started dating and I still lived near DC, so there is a pretty large body of music that would qualify as “stuff we danced to early on and have good memories of” it’s just a matter of thinking of stuff.
MoodIndigo1: that’s always a risk, I guess, but I’ve been to a number of weddings of people who have a dancing background and the choreographed dances looked awesome. We both have a dance background (though mine isn’t in ballroom, as I said, I think I pick up choreography pretty well) so I think we can pull it off.
OpalCat, here’s a data point for you. My niece is a dancer. At her Bat Mitzvah party (age 13, for the Judaism impaired) she and a professional dancer choreographed a number. They performed it as a “first dance”. It was AMAZING!!!11!! The professional dancer/choreographer was also female, and the dance wasn’t romantic, but it was awe-inspiring.
The “regular” dancing my niece and her dance buddies did during the party was also astounding. These kids did some moves that I would have sworn were special effects, except I saw them IRL with my own eyes.
It can work. I think it helps if it looks like you and your groom are having the time of your lives.
My wife and I danced to Motel Matches by Elvis Costello & the Attractions. We wanted something short and kind of edgy.
My wife and I danced to Nick Cave and the Bad Seeds Into My Arms, the song is really beautiful and it’s still not a typical song.
Since you’re both good at it, how about something completely different like an Elizabethan set dance a la Shakespeare in Love? You and your husband lead the bride’s party and groomsmen. Of course you’ll have to practice first, which should be a wonderful way of breaking the ice between the two parties. And if it’s a simple one, having demonstrated it once, everyone else can join in for a repeat.
My favorites - but I don’t think they are what you want…
Beatles, When I’m 64
You’re Timeless to Me, from the Hairspray Soundtrack
I think we danced to “Kiss The Girl” from The Little Mermaid.
Nashiitashii and I are using “Alice” by Tom Waits.
In Your Eyes, by Peter Gabriel might work. It’s a slower song, but not what I’d characterize as a “slow dance” song, with some interesting parts (depending on which version you get) that might make for a cool wedding dance, maybe.
Yes! That one (In Your Eyes) is on the playlist of possibilities I have been building in Winamp. I lurve that song. It’s definitely on the “maybe” list.
Another place to look for possibilities - there are lists on Wikipedia of the music used in Dancing with the Stars for seasons 4 through 7. You’d have to wade through some dreck, and some of the choices are questionable anyway, but you might notice a song that you love on the list.
Oh good idea! I should look for lists from So You Think You Can Dance, too…
How about “Across the Universe” by the Beatles, or the good straightforward cover of the same by Rufus Wainwright?
We danced to Perfect Day, by Lou Reed.
And it was, you know, a perfect day 
This is off base, but…
My wife and I had our first dance to At Last by Etta James. I know, I know – very boring and traditional. But it’s an amazing song, and for those of us who got married a little later in life with a bit of experience under our belt and a sense that we had finally frickin found the right one…well, this song is for us. 
Of course, we didn’t have Beyonce, but we made do…
I always thought the Safety Dance lends itself exceptionally well to choreography (watch Biodome for some inspiration)…