"What month was my Jesus born in?
Last Month Of The Year.
...
You got your January (January),
February (February)
...
June and July
...
On the 25th Day of December
In The Last Month Of The Year
etc."

I think, this song mentions all the months

Now, Tom Lehrer is hardly "traditional," but I think it's worth mentioning his "Hunter's Song:"
"Well, I always will remember,
'Twas a year ago November,
On a morning bright and clear
I went out to shoot some deer...

Arge