Eartha Kitt's "Uskudar" - a Turkish folksong from the 18th century. She probably got it from a recording by Zeki Muren which had been popular in Turkey a few years before.

I have heard the same tune with Greek words on a collection of field recordings from Crete. The tune doesn't sound either very Greek or very Turkish to me, it wouldn't surprise me if it were originally Egyptian or Palestinian.

A more recent one - "Gallows Pole" by, um, Robert Plant? A version of Macpherson's Farewell. And from the 1990s, wasn't GRiD's "Swamp Thing" based on a real old-time banjo tune?