For me, this is easy. Ride nearly 10 miles down a 10-mile dirt road. About 100 yards short of the intersecting road you want to turn onto and continue your travels, there will be a giant, rutted, 2-foot deep mud hole wallowed out by kids playing in their "mud trucks" (after heavy rains) that can only be negotiated by trying to ride straight through it. Your other choice is to turn around and ride back nearly 10 miles and find another route.
I used to like mud, but I don't any more. Call me a pansy if you will. I ride alone and a big mudhole can get you in a lot of trouble real fast. The KLR's not easy to right and get out of a mudhole if you can't make it through and stall in the middle and you're by yourself. I know this from experience. It's taken me up to three hours to extract myself from a mudhole when there was nobody else there to help.
I see what you're getting at, though, not specifically "mud," but offroad in general. I think Spec provided a valuable resource. When I was in the Army, a few times I was stationed where my only off-road option was a "motocross park." These usually entailed some guy who sat by the side of the entrance in a lawn chair drinking beer and demanding money and handing out some kind of waiver you had to sign saying he wasn't responsible if you were injured.
I rode them, anyway. My experiences in these "parks" consisted of wallowing down gnarly, muddy trails while kids on 2-stroke dirt bikes flew by and laughed at me. But, hey, it was better than nothing.
Do you ride alone or do you have somebody to ride with? Cell phone coverage in the area in the case of a breakdown, accident or emergency?
Not to be the Harbinger Of Doom here (possible you have more off-road experience on other bikes?) but it's something to consider if you're riding off-road by yourself, especially on a big, heavy KLR. I guess you have to kind of push yourself to develop your off-road skills, but heed the voice in your head that says, "Maybe I shouldn't continue." If you're green, stay close to the trailhead until you get some practice.
The fact that you're looking for a place to ride off-road kind of says you don't have any riding buddies there that already know where to go. Maybe you can find some? The enjoyment of riding off-road is increased tenfold if you've got some fellow riders along with you to share the experience and be there if you need help.
I hope you find a good place you can safely ride off-road, dude. Good luck to you in your quest! If you find one, show us some pictures of it!