*Smack My Bitch Up? * 
Why Don’t We Do It In The Road (Beatles).
I once went to a Sunday breakast buffet at a swanky place that had a harpist performing. It was all I could do to resist the temptation to make that song request.
If there’s a shotgun involved in this wedding, Havin’ My Baby might be apropos.
Dunno if this helps, but “our song” (because it happened to be the one playing when I proposed to her) is “Tupelo Honey” by Van Morrison.
When we got married, our recessional song was “It Had To Be You”.
“Absolute Beginners” by David Bowie. For dancing you’d want the single version, as the original soundtrack version is over 8 minutes long and has some abrupt time changes which may make dancing sort of awkward.
Sensitive relations may object to the line “As long as we’re together the rest can go to hell”, but otherwise the lyrics seem to fit what you’re looking for.
Everclear’s “Annabella’s Song” was written as a lullaby, but it’s beautiful, with the violins and the loving tone of the words. I think it could easily be a love song.
Is “Come What May” from the Moulin Rouge Soundtrack too cheesy?
Semisonic’s “FNT” is fun and the lyrics are full of adoration.
Another vote for the Cure’s Lovesong. It’s the ultimate wedding song, singer Robert Smith wrote it as a wedding gift for his wife. If you’d like a more up-tempo version, 311 did a cover with a Hawaiian guitar.
Ooh, another one – kinda corny, but one of our faves that we always dance to at weddings (and we DON’T dance well):
[url=http://www.lyrics007.com/Anne%20Murray%20Lyrics/Could%20I%20Have%20This%20Dance%20Lyrics.html]Anne Murray, “Could I Have This Dance?”
Nice waltz beat too.
Crap. Fixed Link.
If I Could Turn Back Time -Cher
I Hate Myself For Loving You - Joan Jett
Heartbreaker - Pat Benatar
Masochism Tango or I Hold Your Hand In Mine - Tom Lehrer
I Will Survive - Gloria Gaynor
Miss Otis Regrets - Kirsty MacColl and the Pogues
Now the serious suggestions:
We Belong - Pat Benatar
They Don’t Know - Tracey Ullmann
Come Rain or Come Shine - Billie Holiday
Heart - The Pet Shop Boys
These Dreams - Heart
I think it’s highly unfair that no one ever has Aqualung performed at their wedding.
You could be the first, gonzoron. 
If I ever get lucky enough to meet someone I would like to marry, this is the song I would like. Its clearly a love song, but a lot of people think its also quite sad, probably because of the tempo. Anyway, the song is The First Time Ever I Saw Your Face by the Stereophonics (originally by Roberta Flack I believe, but its this version I prefer).
Not a lot of people have heard this version, I think you can download it legally from this site.
Keep it sweet and simple. Go with “Love Me Tender” by Elvis.
Love me tender,
Love me sweet,
Never let me go.
You have made my life complete,
And I love you so.
We used Amy Grant’s cover from the Honeymoon In Vegas soundtrack for the last dance at our wedding reception.
“You Make Me,” by “Weird Al” Yankovic.
Now that’s love. 
You mean the Darth Vader theme from Empire Strikes Back and Return of the Jedi?
Ooh, did he also sing the lyrics? You know:
“We’re the evil galactic empire,
We’re the evil galactic empire.
DON’T ask US to be merciful … with you,
To be merciful … with you,
You pathetic vermin scum;
WE’LL crush YOU like a bug beneath … our heels,
Like a bug beneath … our heels,
A-and then, scrape you off.”
You mentioned Tool. How about Parabol?
So familiar and overwhelmingly warm
This one, this form I hold now.
Embracing you, this reality here,
This one, this form I hold now, so
Wide eyed and hopeful.
Wide eyed and hopefully wild…
Knowing Keenan, it could mean anything (and probably doesn’t mean what it looks like) but it’s still a beautiful song at face value.
Parabola’s also good but not really a first dance kind of song. You could use it as a segue though.
Staind’s So Far Away just came on the radio as I posted my previous entry and considering some of the other bands in your OP, I thought it was worth consideration as well.
This is my life
Its not what it was before
All these feelings I’ve shared
And these are my dreams
That I’d never lived before
Somebody shake me
Cause I, I must be sleeping…
[QUOTE=gonzoron]
[li]Runaway by the Corrs (It was sounding great until the “by candlelight, make love to me through the night” which is a bit too evocative for our relatives)[/li][/QUOTE]
If that’s too much for your relatives, then I guess Closer by Nine Inch Nails wouldn’t work, either.
My all-time favorite love song is Peter Gabriel’s In Your Eyes. Not the really bad remake–that was just nasty. I also like Head Over Feet, the Cranberries’ Dreams, Yellow by Coldplay, and All I Want Is You (U2).
I’m going to add another vote for The Cure’s Lovesong. It’s kind of mesmerizing.
Hmmm…I’d suggest:
Love song for a Vampire (Second it, that is. Good song. And no real “vampire” references that’d put off anyone listening to the lyrics)
Last Man in my Life—Sarah Brightman (This won’t work if it’s a lesbian wedding, though. )
Together—From the anime Robotech soundtrack. (Good luck finding it, though.)
…And Forever—From the Big O soundtrack. (!)
Make Someone Happy—Jimmy Durante
Barefoot—K.D. Lang
I Love You—Sarah McLachlan
The First Time Ever I Saw your Face—Celine Dion (Johnny Cash also does a version, which is…interesting.)
Not that I’ll ever (probably) get married, but if I were to get married and had a reception, my first dance song (which you may use if you want) will be:
Marvin Gaye’s “Let’s Get it On”.