When I am teaching people who like country music, I use "Cry" by Faith Hill for a good waltz. Also, though it is older, "Tennesee Waltz" still is a popular one.
But there are some nice waltz tunes hidden amongst the country music genre, which is not one of my favorites, except for when doing a waltz.
"James" by OAR, for the college set, works for a waltz as well and is a little faster tempo.
And I agree with SmoothGeezer. The slower ones are harder. "Moon River" for example, is considered one of the great waltz tunes, but it is quite slow.
If you want a good Viennese Waltz tune, try "Iris" by Goo Goo Dolls.

Hi,

There are many more modern waltzes out there.

The Eagles - Take it to the limit
Engelbert Humperdink - Funny Face
Some Avril Lavigne songs
Donna Winner - It is you

You'll be surpirzed to know that most new punk rock music are waltzes as well.

Enjoy!

Waltz

A natural Woman Aretha Franklin
Are you lonely tonight Elvis
Rainbow Connection Carpenters
Satellite Dave Matthews Band
If you don't know me by now Harold Melvin & the Blue Notes
Come Away With Me Norah Jones
Be Here to Love Me Norah Jones
Que Sera Sera Pink Martini
She's Leaving Home The Beatles
Take it to the Limit the Eagles
Open Arms Journey
Since I've Been Loving You Led Zeppelin)
Scarborough Fair/Canticle Simon & Garfunkel

Viennese
Piano Man Billy Joel
Kiss from a Rose Seal
Never Tear Us Apart INXS
I Put a Spell on You Creedence Clearwater Revival

As I have been listening to my album collection over the last few months I have been jotting down the titles of those songs that might be good to dance to. Here are the most danceable slow walzes that I have come up with so far. Tempos very and some would be easier to dance to than others but I have ommitted the most extreme. I have yet to actually dance to all but a couple. Expect this list to grow considerably as I still have the majority of my collection to go through.

artist / song title / comment
Alison Krauss / Stay
Alison Krauss / Dream My Dream with You
America / Never Found the Time
Art Garfunkel / Disney Girls
Art Garfunkel / Down in the Willow Garden
Art Garfunkel / Mary Was An Only Child
Art Garfunkel / Wayaya
Art Garfunkel / Barbara Allen
BBM / Naked Flame
Beth Orton / Ted's Walz
Carol King / Way Over Yonder
Carol King / A Natural Woman
Chris Isaak / Blue Spanish Sky
unknown from City Sampler #16 / Angel
Indigo Girls / Cordova
Indigo Girls / Kid Fears
Secret Garden / Nocturne
Secret Garden / Pastorade / difficult?
Secret Garden / Papillon
Secret Garden / Serenade
Secret Garden / Atlanta / difficult?
Wishing Chair / 3 Doors
Seals & Crofts / Funny Little Man / tempo changes
Soraya / Pueblito Viejo
Cake / World of Two
Cake / Sad Songs and Walzes
Adam & Rick Wakeman / If Only / too fast?
Adam & Rick Wakeman / Another Mellow Day / too fast?
Seal / Kiss from a Rose
Black Sabbath / Changes
Black Sabbath / Solitude
Black Sabbath / Fluff / fast?
Aeone / Hands of Love
Aeone / Voice of America

Iceman: Great list! Do you have any for other styles? I am making a list for a reunion of my teenage neighborhood of what you might call "hippy music". Can find some good cha cha, but not much anglo music for tango and salsa.

There is a band named Secret Garden. They are a modern - kind of New Age type band. They have some great slow waltzes.

Secret Garden got mentioned here. Does anybody know of a recording of " You Raise Me Up " that can be danced to.

"You Raise Me Up" as a Waltz?

That would be very funny.

Yes, here are two versions of "You Raise Me Up" as a waltz.

http://www.maestrorecords.com/new2008_cdts158.html

Nice version with a definite beat so suitable for beginners too.

http://www.maestrorecords.com/hisaosudo_cdmmp1002.html

Nice orchestral version, slower and more difficult.

There are dozens of versions of "You Raise me Up" on iTunes. ALL of them are in what sounds to me like duple time, and I can't imagine how it could possibly be arranged in waltz time. Is there more than one song of the same name: surely, we can't be talking about the song made a hits by Westlife & Josh Groban?

Can anyone point me in the direction on an online clip of a waltz version - I'm fascinated!