I have a 2006 Wrangler X model and have had this problem since 6000 miles. It is not the tires or the steering damper. I have replace the steering damper 4 times now and it keeps going bad( might I add 4 different name brands). They call it the death steer shake. It feels like the whole front end is flying apart. The only way to make it stop is to bring the Jeep to a complete stop. This Jeep has been in the repair shop more than it has been driven!!! I am being told that all my bushings are bad and have been bad for a very long time....so on that note I think the dealer knew that they were bad from get go but keep pushing it off because it was under my warranty. I have spend many days at the dealer repair shop fighting to just get this piece of junk safe for the road. So I would start by haveing all the front end bushings checked out and I hope that fixes you issues...GOOD LUCK!
I had this problem and it turned out to be a stabilizer bar on the front suspension. Mine did exactly the same thing. It removed the shimmy, but I am following this up with realignment of the front end and rebalancing the tires just in case.