Finally, found the answer to a question Irena Bell, host of Ottawa's Ukrainian radio program had put out in cyberspace.
She wondered if "You Are My Sunshine" was a Ukrainian song, since there are several Ukrainian versions floating around.
The answer is here!
It's not... according to this site, it's the official state song of Louisiana. But it sure is a favourite with Ukrainians everywhere!
You can catch it on Nash Holos every now and again...