This discussion exactly captures a problem I've had now with TWO Ram 1500's.
First one was a 2014 Crew Cab 4 x 4, Longhorn edition. Hemi, not sure what rear end it had. Dealer worked at it for months, balanced the tires several times, rebuilt the rear end, balanced the driveshaft. The drive shaft balance helped a little but in the end they couldn't figure it out and told me "it's a truck".
Since I bought it used, had brought up the issue before I bought it, and they told me they had fixed it, I negotiated a very good trade for a new 2016 Crew Cab 4 x 4, Ecodiesel, Longhorn edition. This one has the 3.55 rear end. It has developed the exact same issue. It's in the shop now for its second oil change and I have asked them to look into it. They've road force balanced the tires and tell me it's better and that the rest of it is due to the tires (BFG All Terrain TAs, very highly rated tires...) We shall see.
Over all my service visits, I've driven several demos, the last two of which had the 3.21 rear end with Hemi engines. They were smooth as glass.
It really doesn't seem like anybody knows how to tackle this problem. It's annoying at best, and in my opinion if a vehicle is shaking as much as mine is, it can't be good for its long-term durability.
If the tire balance doesn't smooth this thing out, I'll continue to push on it. After the first truck, though, I think the service guys are starting to think I'm a little nuts...
jb