Robby mentioned " John Brown's Body, which uses the same melody that was later used for the Battle Hymn of the Republic."
I have read that Julia Ward Howe wrote the words (all umpteen stanzas) of "Battle Hymn" because she was awakened in the middle of the night in 1860 (?) by a large band of folks marching through the street singing "John Brown's Body." (I guess if you can't get back to sleep you might as well write a song!) I think that's why it was set to that music.
Does anyone know more of the story?

Also, while Lonesome EJ seems to be focusing on songs "of" the Civil War, there are some stirring songs written "about" the Civil War.
On one of The Limelighters' albums they have a medley that has excerpts from these three songs: The First Battalion
Two Brothers
Yes, I See

I don't know when these songs were written or by whom, but they are all powerful songs.

BTW, wasn't "Lorena" sort of the "Lili Marlene" of the Civil War (or the other way around?)? On the Ken Burns PBS series (either the Civil War series or the Songs Of The Civil War program) I believe he says that "Lorena" holds a special place as a War Between The States song, because soldiers on both sides would sing it together across the battlefield at night.

I would also like to mention Longfellow's poem (not the song, because the music came later)

I Heard The Bells On Christmas Day

I contains two verses specifically about the Civil War, because he wrote it during the Civil War, partly as a response to the irony of hearing Christmas bells while the cannon were thundering on Christmas Day.
There are two recently active threads that contain the lyrics and discussions of the poem.

Genie