I've had a few sets of AT tires over the years that have been mentioned -
Nitto Terra Grappler - ran probably 6 or 7 sets of these between my Liberty and LR3. Great tire on and off road, deep mud can be an issue. Rocks never a problem. Did puncture a sidewall once parallel parking once. Love the tires though, great wet traction and low noise. No issues airing down to 15 psi.
Hankook Dynapro ATM - have these on my Liberty now. Good traction on and off road, no punctures, but they are LOUD. Like helicopter landing loud. I did have some alignment issues though that have since been fixed. No issues with punctures or airing down.
Cooper Discoverer AT3 - had these briefly on a 96 Cherokee I was restoring. Quiet, great traction on road (never went off road with that one). No rim protector though and one of the tires had a wave in the sidewall. Didn't seem to impact anything.
Yokohama Geolander G015- had these on my 15 Cherokee. more of an aggressive all season than anything. Great traction in the snow and wet, even when they only had about 10k miles left. Very quiet and nice riding. Off road they'd gum up somewhat quick.
Toyo Open Country AT3 - have these on my Renegade Trailhawk. My favorite AT. Super quiet (I took the stock tires off at 9k miles and these were just as quiet), great traction in rain, snow and ice. Have spent a lot of time on the rocks with them aired down to 13psi and no issues at all. I love them.