My experience with this for what it's worth: I had the exact same symptoms. This problem is no doubt not guarenteed to be the same but...
Bnine identified a worn front track bar bushing for me. After I replaced it, the shake changed but did not go away completely. The root cause of the problem was poorly balanced wheels and the worn bushing just allowed more movement once the shinmmy started. Again, on Bnine's suggestion I ditched the steel wheels for high quality aluminum wheels and quality tires. Now, I've had the jeep up to 125kph on the highway and it feels as solid as any car I've driven. I've read some people have had good results with beads or BBs in the tire for balancing which, if the wheels/tires are the issue, might be a cheaper solution.