Another eternal question would be why the ice cream trucks all use it as their jingle. (At least, all the ones I’ve heard do).
Going back to “Turkey in the Straw” thing - they’re playing T in the S (I originally wrote it as TitS, which looks really wrong), not DYEHL. (cite: Turkey in the Straw - Wikipedia)
Why they do is a good question, though. Sounds like something Dopers should investigate. Has nobody asked Cecil before?
In the movie, Easy Rider a musical troupe is singing it as “Does your hair hang low” in the commune scene. As “hair” is the thing that really makes the most literal sense with the lyrics, I suspect that’s the original version, with “balls” becoming a ribald parody, and “ears” becoming a more bowlderized parody.
They don’t all play Turkey, though… The Entertainer is also a common ice-cream-truck song.
It probably has something to do with, a, being a simple enough tune to reproduce recognizably on a poor-quality sound system, b, being lively and cheerful enough to attract kids’ attention, and c, being in the public domain so the ice cream company doesn’t have to pay royalties for every block they play it on.
I’ve heard Turkey In The Straw (I almost abbreviated it, but then I saw what it spelled) from ice cream trucks, but when I was a kid they played a different tune.
I went to this page and entered it on the keyboard, and it said the notes are these:
e’‘4 e’‘4 d’‘4 d’‘4 c’‘4 e’‘4 g’4 c’‘4 b’4 d’‘4 g’4 c’‘4 c’‘4 e’'4 g’4.
(You can cut’n’paste those into the box on the page.) I didn’t find the song in the first few pages of the search, so I’ve no idea what it is.
If that’s Turkey In The Straw, then someone’s using different tunes than I do. Becuase what they play is what I know as Do Your Ears Hang Low.
I have always assumed that it’s a vestige of the Revolutionary War, much as “stuck a feather in his cap and called it macaroni” is - that these things stick on a lot longer in song than in speech.
Ah, I guess you’re right. Thanks for disabusing me of my mistaken notion.
I wonder whether the “throw them over your shoulder” refers to rifles, as suggested above (it would really be muskets, I presume), or to those white things that criss-cross their chests and go over their shoulders.
Another cultural divide; around here they play Greensleeves. 
Why are people having such a hard time believing that this was invented in the first half of the 20th Century by camp counselors and using the word “ears?”
According to Wiki:
My answer to that is that it is unlikely that the song was in existence as long ago as the Civil War(much less the Revolutionary War) without the words in one form or another being written down and published until the 1940s.
The Wikipedia article says that folklorist Vance Randolph collected the words(supposedly with “balls”) in 1941, and they weren’t published until 1991. But no reliable source is given. I’d sure like some better evidence that it was originally “balls” from the military.
Or ‘hair’, which makes more sense.
Like I said in the OP, ears don’t wobble or hang. Hair does, and yes, people talk about testicles hanging low, too. So it seems more likely to me that it started out as something different.
The fact that “hair” makes so much sense is, I think, a point against it. It’s not at all uncommon to be able to tie hair in a knot, and when it is possible, it’s readily apparent. Why write a silly song about something so obvious?
samclem, one might well turn your question around: Why do you have such a hard time believing it originated in WWII with “balls”?
Some years ago, I posted a similar question as the OP: “Like a Continental Soldier”.
I was mainly interested in how (or if) the act of throwing one’s ears/balls/hair/whatever over one’s shoulders was derived from the dress of Continental soldiers, but there’s some discussion there about the original lyrics. We couldn’t arrive at a consensus regarding any of these issues, however.
As for the “hair” theory–it’s true that eighteenth-century men wore powdered wigs with long ponytails. However, the ponytail hung straight down the back–not flung over the shoulder. Plus, I think men still wore their actual hair pretty short (so the wig would fit more comfortably). I’m inclined to regard the “balls” interpretation as the most likely one, but haven’t seen any conclusive evidence.
If it was “balls” it seems to me that the author must have had a pretty warped imagination - tieing your testicles in a knot or bow seems pretty strange to me!
Jeeze, not to me. Maybe you balls are way too inflexible? 
Well, I"ve found a better look at the story about Vance Johnson, the folklorist, collecting this low these many years ago. And I now accept that the original version may have been “balls,” although whether it came from the military is still open for discussion.
Here’s a link to a recent book in which his references are cited rather nicely.
Or Music Box Dancer.
Si