You're thinking of the song that's usually known as I'll Tell my Ma, and I think it's in the DT database under that name. It's an Belfast children's street song and was recorded by the Clancy Brothers (and no doubt others). It's not the same tune as the cabbage song, though, it's a bit more tuneful, but it's still a simple tune and I guess three chords would do it.

I think it was the boys who wouldn't leave the girls alone when that recording was made, but hey, that was about 40 years ago, maybe things have changed.

Other easy ones might be Banks of the Ohio and On Top of Old Smoky.

Good luck,
Marje