"I love Greensleaves, although I never really thought of it as a Christmas song."

The "Christmas" version is "What Child Is This" (to the tune of Greensleeves)... It's one I like, though not as much as the Coventry Carol.

How about "The Boar's Head"? (I've actually done that, at an SCA feast, while processing in with a platter with an actual Boar's Head, apple in its mouth and all...)