I also remember this song from elementary school in Northern Ohio. That would have been in the late 1960's. Parts of it are stuck in my head and I came on line looking for all of the words. Seems I'm not the only one. This is all that I remember:

Down by the river mama washes my shirt
She has to use a stone to knock out the dirt
The river is shallow, the kids run about
And when its time for supper, mama has to pull them out.

don't go away, don't go 'way mister river
can't you stop and play as you travel out to sea?
don't go away, don't go 'way mister river
can't you stop and play as you travel out to sea?

Down by the river there's a good fishing hole
Down by the river cut a big bamboo pole
-------------when he can be free
He likes to------'cause he likes to fish with me.