Barbara,
There are certain songs that really call for acapella choral singing.

Bright Morning Stars is an example from the Ozarks. Sandy has mentioned shape-note hymns. Wild Mountain Thyme of course but there is a world of choral folk music out there. Check out Pete Seeger's wonderful little record of "The Songswappers" that have South African Zulu rounds such as Soma Gwaza, Bayeza, Wimoweh and more. There are Bulgarian tunes, (Theodora is Dozing) from Phillip Koutev's Music From Bulgaria, Russian folk tunes (Pianitsky Chorus or Red Army Chorus for example) and many Slavic tunes. Indian and Asian tunes lend themselves to unison singing acapella quite well. South Pacific tunes such as Deep Blue Sea and Bahamian and Carribean tunes lend themselves well. Check out Olive Walker (Tie Me Coverlet Down) if you can still find this record in print.

Here's some to consider.

Bright Morning Stars
Steal Away
Most any Sea Chantey
Chain Gang Songs
Gospel Quartet Songs (Golden Gate Quartet...Pilgrim Travelers etc.)
Kumbaya is good for choral singing
Amazing Grace

If I think of more, I'll submit 'em.

Frank Hamilton