I thought this whole thread was a big leg-pull, so I checked it out by listening to as many sound samples as I could find on CDNOW. The great majority of them were instrumentals. But then there was this version by the Holy Modal Rounders:

Flop-eared mule, flop-eared mule, flop-eared, flop-eared, flop-eared mule.
Flop-eared mule, flop-eared mule, flop-eared, flop-eared mule.
Mule, mule, mule, mule, flop-eared, flop-eared, flop-eared mule.
Mule, mule, mule, mule, flop-eared, flop-eared mule.

And this version by a band called Doodoo Wah, on a CD called "For Amusement Only," Ballum Rancum CD #4, 1996:

Corned-beef hash, corned-beef hash, corned-beef, corned-beef, corned-beef hash.
Corned-beef hash, corned-beef hash, corned-beef, corned-beef hash.
Hash, hash, hash, hash, corned-beef, corned-beef, corned-beef hash.
Hash, hash, hash, hash, corned-beef, corned-beef hash.

For those who want to compose their own verses, it seems the rule is: You need a 3-syllable phrase, of which the first 2 syllables are an adjective or adjective phrase, and the 3rd syllable is a noun.

Only Andy Griffith seems to sing "real" words, but then he has a different tune also (on "Favorite Old-Time Songs," Madacy CD #3130, 2000):

Took my wife to the barnyard and I set her down to supper.
Well, she got choked on a turkey leg and stuck her nose in the butter,
Stuck her nose in the butter, stuck her nose in the butter.
Well, she got choked on a turkey leg and stuck her nose in the butter.
Whoa, mule, you kickin' mule. Whoa, mule, I say.
Tie a knot in that old mule's tail 'fore he runs away.

I don't know where his version comes from, but it's not the same as Kickin' Mule either.