Oh good, another person here who knows who she is. That makes, what, six? 
It’s a great song, and a real tearjerker for me, personally.
“Beautiful” by Gordon Lightfoot.
John Denver’s performance of the song “Junk” is very simple, and might fit your bill. (It was written by Paul McCartney.)
Er, you can listen to a sample of “Junk” here at Amazon.com to see if it fits your bill. Also try Paul Simon’s “Flowers Never Bend With The Rainfall” for size. Geoffry Oryema 's Makambo is breathtaking IMO. For variety I’ll mention Paula Cole’s “Hush Hush Hush”, it is voice, piano and flute, as well as Jane Siberry’s “It Can’t Rain All the Time” from the Original Crow soundtrack, both are beautiful and simple.
Richard Thompson has written and performed a few:
1952 Vincent Black Lightning
King of Bohemia
How Will I Ever be Simple Again?
And the whole album ‘Small Town Romance’ (if you can get it - it’s very rare).
Why ‘Amazing Grace’ or course.
Recorded music has made too many people about singing just for the fun or it. Go ahead, belt out a tune now and again.
Paul Simon, “Peace Like a River”
Lyle Lovett does a cover of “Friend of the Devil” on a Grateful Dead tribute album that I just love.
Nirvana’s unplugged version of “The Man Who Fell To Earth”…(I think it’s the only version.)
Allison Krauss: “Baby Now That I’ve Found You”
Simon & Garfunkle: “For Emily (Wherever I May Find Her)”
Thanks for the tip. My father sent on a cassette of their stuff quite a while ago and I hadn’t listened to it yet. This makes me want to rush home and play it. 
My votes? “Catch the Wind” by Donovan. And “Cold Tea Blues” by Cowboy Junkies.
Wonderful Tonight, by Eric Clapton.
I’ve always considered this one of the best love songs of the rock era.
Some obscure ones:
Darling If by Spirit
Lullabye by Emitt Rhodes
Most of John Lennon’s album, Plastic Ono Band, is John and an instrument (usually piano). All of the tracks have very simple and spare arrangements. Some songs are very raw (“Mother”) or simply beautiful (“Love”). Great album.
Linger - The Cranberries
Van Morrison’s “And It Stoned Me” on his Moondance album.
Creedence Clearwater’s “Have You Ever Seen the Rain?” and “Who’ll Stop the Rain?”
Simon and Garfunkel’s “April, Come She Will.”
Hank Williams’ “Mansion on the Hill,” “I’m So Lonesome I Could Cry,” and “Lost Highway.”
Iron Hand by Dire Straits
And his “Promises”.
Gordon Lightfoot has been mentioned, but Sarah McLachlan’s cover version of Gord’s beautiful Song for a Winter’s Night is well worth tracking down.
My favorite band, Cowboy Junkies.
Their first album The Trinity Sessions is all in this format. Some of their other stuff is more complex and more rock, but they have plenty of simple songs and these are my favorite ones.
Yes.
“Handbags and Gladrags” by Rod Stewart.
The lyrics make zero sense, but it’s a heartbreakingly lovely song.