Good advise so far!
I had a similar problem this spring with my Peterbilt motorhome conversion project. Here are two additional items to check:
1> Make sure the steering box, steering shaft, and all the drag-links and tie rods are tight and operate smoothly with excessive end play. If you have excessive "play" in the steering wheel when you mive it side to side, this could be a problem. I had a loose fit between the steering box output shaft and the pitman arm. Tightening it up with an impact eliminated all the play. I did not have to replace any parts. But, I keep my eye on it!
2> My real shimmy problem was most evident at speeds between 45-55 mph; it was speed sensitive. For this, I went to my local Peterbilt dealer, whom I do regular business with. The mechanic jacked up the front end, one side at a time, and place a 4x6 block of wood about 24" long up against the tread of each front tire to check the run-out. One side was fine, but the other side revealed a "bulge" in the tire, making it about 1/2" out of round.
As I needed new steer tires anyway, I replaced both tires with nice new Goodyears, and the problem was solved!
Hope this helps.
Speed Gray, K8SG
Grand Rapids, MI
Highway OPie (Speed Gray) Grand Rapids, MI: 2003 Peterbilt 379 Motorhome; 550 hp 6NZ Caterpillar C15, Eaton-Fuller 18 speed transmission, 3.36 rear; 63" sleeper and 16' Morgan box.(firstname.lastname@example.org). There are many Peterbilts, but this one is mine!