The patriot's 4x4/awd system is very good but tires are what makes the vehicle get through what it can. Since I've used 7 different tires on my vehicles (A patriot and Compass) i'll chime in with some real world advice. I live in an area that gets about 5-6 months of snow a year. An all season tire is crap in the snow, with the AWD it will barely get your by. Example, Michelin latitude HP all season tires, great tire but junk in the snow, with 2-3" of snow on the ground and a slight incline, I couldn't make it up the hill. The stock goodyear SRA tires, barely passable as a tire i'd use in the snow, even though rated as an AT tire it was a street tire.
I have to disagree with Ignatz that AT tires are loud. I've used both the firestone destination AT and Cooper AT3 in the winter. Both are an AT tire and both were no louder than any of the street tires I've ran on my vehicles. If you get into the more aggressive MT tires you start to get loud (my 15" wrangler territotiy tires are not loud but aren't quiet either, you know they are there).
I also ran a set of dedicated snow tires one winter Gislaved Nord Frost tires. The tread pattern wasn't the most aggressive for offroad, but they were fantastic in all conditions and only struggled in deep snow, but were still better than street tires.
My criteria for a tire is good in snow/offroad and not too loud. The firestone Destination AT and cooper AT3 are probably what you are looking for. Between the 2 the only difference I found is that the firestone was ever so slightly better in the snow and ice. Both were quiet and performed well offroad as well.
My wife has a Cherokee KL and it has crappy continental street tires, its 4x4 system is superior to my compass...........and I've gotten it stuck in snow numerous times only due to the tires.
Take my advice for what its worth, The Cooper's or firestone are great tires, and I've heard good things about the Yokohama geolander AT/S I just don't have personal experience with them. With an AT you make some compromises, but less so than an all season tire or strictly winter tire.