The truck will shake due to balancing which will come out of wack often. When is the last time you had your truck balanced? Given that you have it happen in the 60MPH range I bet you its the balancing on the tires.
My Impala maybe 2 times a year would need to get balanced because of the crappy roads in NJ and the shimmy shake would just eat away at me. Tell me this, at 60MPH, if you take your hands off the wheel(not fully off but light so you can still grab it) does the wheel shimmy/shake? If it does, take it for the balancing, its usually like 5-10 bucks a tire and takes like 10-15 minutes.
Balancing could be from tire wear, a weight on the rim falling off etc. Its a cheap fix and worth it if you haven't had it done recently.
2004 Nissan Xterra XE V6
+Front/Rear Bilstein Shocks
+Bilstein Steering Stabilizer
+Kumho Road Venture APT 265/70R16
+Intake Modification + OE/Filter