I'm so excited to have finally found this. I guess I was having a hard time finding it because I was searching for slightly different lyrics.

This is how I learned it from my mother. (She was born in 1945 and learned it from her mother.) They were both born and raised in New York. I dreamt about this song one night, only to find out the next day that my grandmother had passed away.

I open up my pigeon coop
and let all the pigeons free.
They fly all around on every bough
and land on the tallest tree.
And when they return from their merry, merry flight
they shut their eyes and say good night.
Coo-roo-roo-roo-roo
Coo-roo-roo-roo-roo
Coo-roo-roo-roo-roo, roo-roo-roo-roo