Hello Marc,
Londonderry Aire (Danny Boy) is on my CD "Puttin' on the Ritz" on Nagel-Heyer records. It has been recorded many times by others as well. Bill Frisell has a "Best Of" CD on Nonesuch that collected several of his wonderful tracks of folk songs. Charlie Haden and Hank Jones recorded "Steal Away". Monty Alexander and many others recorded "You Are My Sunshine" and " Battle Hymn of the Republic". Monty also performs music from the Caribbean as do others including the Caribbean Jazz project. Toshiko Akiyoshi has written several charts that are influenced by her native Japanese music including "Tanuki's Night Out". Randy Weston, Hugh Masekela, Miriam Makeba, Abdullah Ibrahim, and many others have incorporated a great deal of music of Africa in their music. There are also numerous Afro-Cuban groups. Jazz has truly become an international music. I invite others to contribute some specific songs.
Saxophonist Ron Blake contributed "Christmas Comin'", a folkish tune from his home country, Virgin Islands, on a holiday album we did many years ago. We did it a la "St Thomas".
Just remembered an album released in 1964 by Art Farmer Quartet on Atlantic called "To Sweden with Love" - all jazz arrangements of Swedish folk songs. Steve Swallow is on bass, Pete La Roca on drums and Jim Hall on guitar.