How heavy is heavy? A regular length bike can handle quite a lot of cargo if the load is balanced well. It's not unheard of for touring bikes to carry more than 70 lbs. A bike that isn't designed around cargo will usually max out somewhere in the 40-50lbs range.
The real problem with a regular length bike is it doesn't handle bulky cargo well. Building materials and the like are *right* out. An Extracycle or other long-bike design will handle some kinds of bulky cargo. If you're getting into real weight (say, over 200 lbs), there probably aren't any bikes that will do the job alone. And really, at that kind of weight, most trailers won't do the job and I'm not sure off road is the brightest idea either.
For normal off road use (no jumping tree trunks etc), any reasonably sturdy bike should do the job. A rigid bike will handle a lot more in the way of potholes, speed bumps and tree roots than you might expect
. As long as it's on gravel roads, fire trails etc, you'll probably be ok. If you anticipate things like riding down stairs or jumping downed trees, a bike is probably not the way you want to haul cargo.
Often on flat ground, I need to gear down just to get the bike moving with a heavy load. If you haven't done a lot of loaded biking, you may be overestimating the repair problems of a geared bike and underestimating how hard it is to move stuff around on a bike. I can count on one hand the number of problems I've had with my drivetrain over the last year, and there's no way I can count how many times I needed a lower or higher gear than my normal cruising speed. (sum total of drivetrain problems: chain dropped twice. first time, I couldn't fix it myself and took it to the LBS for a quick lesson. second time, I fixed it myself and continued on my merry way.)