Marching Through Georgia
Kingdom Coming
Bonnie Blue Flag
Battle Cry of Freedom (there was a Southern and a Northern version)
Poor Soldier
The terminally Un-PC Dixie (actually written before the war; one of Abe Lincoln's favorite songs)
Southern Soldier Boy
Of course, I'll second the motion on Lorena
Hard Tack (parody of Hard Times)
Cheer Boys Cheer
The Homespun Dress
Shiloh's Hill
Virginia's Bloody Soil
Roll Alabama Roll
The Vacant Chair
I Goes to Fight Mit Sigel(parody making fun of German soldiers, probably not PC)
The Invalid Corps
The Army Bean

Songs that predate 1861 but were still being sung:
Annie Laurie
Listen to the Mockingbird
Hard Times
Home Sweet Home
Poor Wayfaring Stranger
Shenandoah
Johnny Is Gone For a Soldier
Minstrel Boy
Endearing Young Charms
Sweet Betsy from Pike
Auld Lang Syne

Hymns:
Amazing Grace, the only hymn ever written
Rock of Ages
On Jordan's Stormy Banks
Stand Up, Stand Up for Jesus
Nearer, My God to Thee
There Is a Fountain Filled With Blood

My favorite post-war nostalgia song: I'm a Good Ol' Rebel. I'm not sure but I think Faded Coat of Blue falls into this category also.