I don't have any new rhymes to add, but I'd like to give you the versions of a couple that I grew up with, in Southern California in the late 80's and early 90's:

Down by the banks of the Hanky Panky
Where the bullfrogs jump (sometimes we said "croak" instead) from bank to banky
With an eep, opp, soda-pop
With an eep, opp...ker-plop!

I was also surprised by the word differences in this one from what you put down (I'm not sure if you heard it as a rhyme or as a song. Ours had a definite melody, though- pretty whiney and screechy):

Great green globs of greasy grimey gopher guts
Mutilated monkey meat
Itty-bitty birdy feet
All wrapped up in a package full of porpoise pus
Swimming in a pool of blood

My mom also taught me one from her childhood (1960's southern California):

Acka-backa soda-cracka
Acka-backa-boo
Acka-backa soda-cracka
I see you! (it might've been "out goes you," now that I think of it)

demographics for mine are pretty much all white kids. The frog one was just girls, whereas everybody did the gopher guts one. Not sure about the soda cracker one, since I wasn't there.