kchaggins ... I have driven a '99 Silverado LT extended cab 5.3 3.42 Z-71 for 14 years and 300,000 miles... still own it and still rides like a dream. Replaced it with a new 2014 Silverado LT Double Cab 5.3 3.08 gears Z-71 and wish I never did. This truck shimmies, shakes, vibrates all the issues you see on these post. GM has "looked" at it more than several times, and found it to be 'driving as designed'. IF this is true, GM design engineers should be fired.
As a side note, I am starting to think some of the vibrations are coming from the transfer case. When switching into 4 Wheel Auto (at high speeds), there is gear noise and a high pitched "whining" that is very noticeable.
Either way, this could be an excellent truck. The vibrations just need to be figured out!