Lynn, thankyou thankyou thankyou thankyou – you've saved my one remaining brain cell from blowing itself out trying to remember the rest of that icka-backa-soda-cracker rhyme. We used it as a jump-rope chant (central California, 50's) and you had to jump out of the rope while it kept turning, on the same rhythmic beat that the next kid in line jumped in.   

I first picked up eeny-meeny from the bigger kids in my street when I was four. They used the N word, and I didn't know what it meant. When my mother heard me reciting it she told me that wasn't a nice word to use and that I should sing "tiger" instead. I subsequently heard the chant used both ways, so there were two versions floating around.

In the 60s we had a variant of that "beestay" rhyme-&-response (I never knew what a Beestay was either) which I first heard at a high school football rally and everyone thought was way cool. Ours started with

flea (flea)
fly (fly)
flea fly flew (ditto)
coomalata coomalata coomalata beestay
no no no no not the beestay

and ended in a sort of scat-rhythm: eee-biddlety-oaten-doaten-wahbat-skee-watten-tatten-SHHHHHHHHHHHHH !!!!

Kitty, we also had the same "one-potato-two-potato-three-potato-FOUR; five-potato-six-potato-seven-potato-MORE" and whoever was More was out, or It, or whatever. Interesting how such similar/identical things pop up on both sides of the Atlantic.

Do kids these days still sing these rhymes? Or is it all computers & video games now? Can any playground supervisors or parents of young kids shed any light?