When I was in elementary school (early 2000s, Los Angeles) we sang it:
Down by the banks of the Hanky Panky
Where the bullfrogs jump from bank to banky
Where the eeps, ops, soda pops,
Froggy missed a lilly and he went ker-plop!

I don't remember what game went with it. I do remember it having a tune:
F, F, F, A, A, A, A sharp, A sharp, C, C
C, C, F,F, A,A, A sharp, A sharp, C, C,
C, C, F (octave up from the other ones), C, A, C, F
C,C, C, C, C, D, C, A sharp, A, G, F
I thought I should include the tune because I'm not sure how widespread it is. Some of the versions here don't seem like they would fit it.