Mystery vibration at 25 mph can't be diagnosed. It's not the front end!

I have a 1965 dart 270 w/ slant 6 and I'm getting a body shutter upon acceleration at 25 mph. The mechanics are scratching their heads. It doesn't appear to be the front end steering linkage and suspension. It's in the body or drive line. They checked the drive shaft and U joints and they seem to be okay. It's been in the shop 10 days and they can not find the source of the mystery shutter. I had the tires rotated and had alignment. All tires are at a proper psi and balanced. Does anyone have some clues to fix the problem. It Just started doing this a couple of months ago. The problem just came out of nowhere.