If this Dylan is too un-hip for you, I do have another tip. Scour the Internet and you should be able to find and download a copy of the two-hour Christmas special Dylan did in 2006 for his great (and apparently now defunct) XM-Sirius satellite program, Theme Time Radio Hour with Bob Dylan. There, you can hear Dylan offer witty introductions to all sorts of obscure Christmas blues and country records—along with Brave Combo's polka version of "Must Be Christmas," which he borrowed for his own version, all the way down to including the names of recent presidents when it comes time to recite reindeer monikers.
Dylan's sincere take on "Little Drummer Boy" is on the jump.
[Hat Tip: @ChrisWillman]