Thank you everyone for this thread. Here is the version I learned from my dad in the '40's. He obviously learned it when he was going to school in a small Kansas town in the 20's.

'Way down yonder on Beaver Creek
Sing song kitty cantcha ki me oh
Darkies grow to be ten feet
Sing song kitty cantcha ki me oh
CHORUS:
Kee mo Ki mo Dee ro art
Me hee, Me hi, Me hum-drum pennywinkle
Tit tat pitty pat, blue-eyed "poosy" cat
Sing song kitty cantcha ki me oh.

Then, a few years later, I learned ANOTHER version of the Chorus from the traveling music teacher who came to my elementary school about once a week or so (this was also in the 40's):

Ki mo kairo, dearie-oh mah-ware
Me hi, Me hee, Me instep Sally Sinkle
Sometimes pennywinkle
Instep step back
Hit 'im with a brickbat
Sing-song kitty cantcha Ki me oh.

Would live to hear from anyone who remembers these versions.