Which reminds me, in our ceremony we had “One Hand, One Heart” from West Side Story, which was my pick. 
Check out Dear Life by Anthony Hamilton. Great lyrics, and some awesome guitar.
I really wish you would do “The Hungry Wolf” by X. Sure, there’s a whole bunch of stuff in it about, well, wolves hunting down their prey, and maybe a wedding’s not the best time for a song whose lyrics include “the altar of your death.” But at its heart, it’s a song about two individuals who have chosen to spend their lives taking on the big, bad world together. I think there’s a certain romance to that. It also rocks pretty hard.
OK, totally legit suggestion here: “I’ve Been Waiting” by Matthew Sweet.
You Are The Best Thing - Ray LaMontagne was our wedding song
It’s got a great and strong melody, so it should be pretty adaptable to either piano or guitar. The tricky bit is pulling off that instrumental break and making it sound appropriate, IMHO. There’s some reasonable attempts at covers of it on YouTube, if you’re looking for inspiration.
First dance at my wedding (sorry, would prefer to link to original–could not find it, just some ok covers) was Bruce Cockburn’s Someone I Used to Love. The title is quite misleading:
All the days we’ve been together
All the days we’ve been apart
Add up to a bunch of nothing
If I’m not still in your heart…
and…
*May no shadow ever fall,
That will make me have to call
You someone I used to love
*
Hope that’s not too much, Mods.
You Are My Flower - Flatt & Scruggs
“Can You Please Crawl Out Your Window”
j/k
Serious answer: “If I Needed You” by TVZ. Like most of his songs, it’s simple, elegant, and profound. Link: Townes van Zandt - If I Needed You - YouTube
You can also see versions on YouTube by a younger Townes, as well as the incomparable Emmylou Harris. In my search though, this one best illustrates both the phrasing and unaccompanied guitar.
Kate Miller-Heidke - Space They Cannot Touch (Fits in well with the Ben Folds thing - as well as being a similar style of music, she has toured with him extensively and sing on his latest album, duelling on “From Above” and backing vocals on “Saskia Hamilton”.
My favourite place is you and me.
Uh, wow. My circle of friends have this song picked out to play… AT OUR FUNERALS. We’ve all talked about how it’s a perfect atheist hymn about death and dealing with it.
Everybody gets a free pass on crying at their own wedding. If you can’t cry then, when can you?
And thanks for mentioning this. It reminded me of “Cause I Love You” another Johnny Cash song I always thought would be great for a wedding but hadn’t heard in ages and had actually kind of forgotten about.
In My Life - Beatles
Always - traditional
Can’t Help Falling in Love - Elvis
As Time Goes By - ? Humphrey Bogart?
Got A Feelin’ For Ya by Kelly Willis
Hope For Us All by Nick Lowe
Dooley Wilson. Bogart was the star of Casablanca, Wilson was the sidekick.
My suggestion would be “Would You Lay with Me (In a Field of Stone)” by Tanya Tucker.
“Give yourself to Love” by Kate Wolf Kate Wolf's Song - Give Yourself To Love — Official Kate Wolf Website