I was talking with my S.O. and we were talking about this. Anybody have any?
So far, my ideas haven’t been the best:
Don’t Go-Hothouse Flowers
Slide-Gus
So Long, Farewell from the Sound of Music
“Goodbye, Good-Lookin’”- Robbie Fulks
“The earth sure felt like heaven…for awhile”
Unanswered Prayers by Garth Brooks http://www.planetgarth.com/lyrics/unanswered_prayers.shtml
She’s Finally over me - by Teddy Pendergrass, can’t find a link.
“Don’t Think Twice, It’s Alright” by Bob Dylan.
Probably not what the OP was referring to, but after a particularly drawn out and ugly breakup, it was particularly uplifting for me:
“I ain’t sayin’ you treated me unkind/ you coulda done better but I don’t mind/ You just kinda wasted my precious time/ but don’t think twice, it’s alright.”
“Love Lives On” by Joe Cocker…also, in a slightly different vein…“Letting Go” by Joe Cocker. Yes, I am on a tangent these days, why do you ask? 
“The Dance” by Garth Brooks
What Might Have Been by Little Texas
Don’t Happen Twice by Kenny Chesney
(Although those two are more “looking back after some time has passed” rather than “saying good-bye” songs)
A Girl’s Gotta Do What a Girl’s Gotta Do by Mindy McCready
Oh, here’s The Dance, if you’re interested.
Oh, wait. Damn. Can’t believe I forgot this one. A Better Man by Clint Black
Losin’ You (Might be the Best Thing Yet) by Jimmy Martin - Edria Humphrey.
(Imagine the fiddle here…)
"The longest day I live I’ll still be suff’rin’ from the shame.
In the gutter you were makin’ yourself quite a name.
…
I wanted you, you wanted anyone you could get.
Losin’ you could be the best thing yet."
Damn! For ‘could’, read ‘might’.
I even read it over in preview. Maybe I shouldn’t have been singing along. 
Leaving on a jet plane. I don’t know who sings it for real, I have a cover version by Me First and the Gimme Gimmes.
I believe it was John Denver. [Canadian artist plug]Chantal Kreviazuk[/plug] also sings it excellently, as featured in Armageddon.