My Grandmother use to sing this to all her grandchildren. She was born in 1901, and lived on the north fork of Long Island, NY until the mid-50's. That area was a melting pot of many cultures: Greek, Russian, Italian, Dutch, English to name a few. The words are just slightly different, as I remember:
My pigeon house I open wide
and set all the pigeons free.
They fly through the air on ev'ry height
and light on the tallest tree.
And when they return from their merry merry flight
I close the door and I say goodnight
Coo-a-roo, coo-a-roo, coo-a-roo, coo-a-roo
coo-a-roo, coo-a-roo, coo-a-roo.