I learned this in western Tennessee about 1969 or so as a folk song (along with Goober Peas). This is as I remember it:

Down by the river mama she washed my shirts.
She liked to use a stone to knock out the dirt.
The river is shallow, the kids jump about
And when it's time for supper mama has to pull them out.

Don't go 'way, don't go 'way mister river.
Can't you stop and play as you travel out to sea?

Down by the river, got a good bamboo pole.
Down by the river, got a good fishin' hole.
When papa gets time off, so he can be free,
He comes down by the river and goes fishin' with me.

Don't go 'way, don't go 'way mister river.
Can't you stop and play as you travel out to sea?