I'm out pretty often in the wilderness but the world looks different in a fullsize truck than it does in something compact like an XJ or a side-by-side. Most of where I used to travel in compact 4x4's without a problem would do nothing but crunch metal on my Ram.
From Mesa the closest place to you is probably up at Butcher Jones, trouble with that area of course is it's just a bunch of overused, braided dirt trails that go back to the coves where people make a mess, throw their trash and try their hardest to get the place closed down permanently.
If you're up for a bit more adventure you can travel a bit farther; Rug Road is a hell of a slog and will definitely put a power wagon through its paces but shouldn't be a guarantee of body damage if you're attentive and practice proper technique.
Another fun route, closer to Mesa is take Happy Camp Canyon rd. off US60 before you get to Superior, ride that up to Queen Creek Canyon Rd. and exit on Cerro Road (because the exit route along Queen Creek Canyon rd is gated and I would like to offer you assurance that driving a Ram around the upper gate is not likely to end well for you in addition to being trespassing) . The climb out on Cerro Rd. from that gate is rugged but I've done it in my 1500, you can do it in a Power Wagon. Couple really nice campsites out there too and as close to zero traffic as you can hope for that close to Phoenix.
