I will just throw out my current experience.
I've had a shimmy and a pull to the right and I can feel it at my steering wheel, in my 4Runner.
I had a low speed shimmy and a high speed shimmy. Worse when braking.
Today I had my tires rotated with an oil change. The pull to the right stopped. And the shimmy stopped.
I am suspecting that my tires are Old. And there is tread separation occurring.
~~~~
This very same thing just happened to me in my 1976 Cadillac El Dorado. The shimmy was so bad, that on cold mornings, that the car would shake side to side! The factory code numbers on the old tires could have put the old ones in the mid 1980's mfg. point in time! Yet they looked like new, with no tread wear.....YIKES!
I put brand new Coopers on the El Dorado in Feb. 2010 and the shimmy stopped. And the pull to the right stopped. I've had more than one shop tell me this is tread separation.
Sometimes, it can be the tires. Check the MFG. Date Code on the Sidewall.
.