There's endless discussion online about what tires work best for dual-sport. Of course, what works best for you depends on your bike, your riding skills and style, terrain--and, ultimately--budget.
I'm a bit of a cheap bastard for my KLR (hey, it's about having fun--not worrying endlessly about keeping chrome polished!), so I'm not going to mount rubber that's a sizeable proportion of the value of my bike. And the reality is, family and work commitments restrict me to being mainly a weekend warrior. So my choice of tires may not be the "best" in terms of available performance, but overall I've found them to be a good compromise for gravel roads, the occasional rock garden, and muddy ATV trails.
I've settled on a Kenda K270 for the rear and a Pirelli MT21 for the front, and heavy-duty inner tubes in both. I ordered my tires from www.aviciouscycle.ca where I've had great service. I had K270 on the front and rear previously, and while the rear is fine I have to say the front is pretty lousy on fire roads. The MT21 really bites into the dirt, improves control on rutted terrain (the bike actually goes where you point it) and gives a much more predictable wash-out in cornering. The rear is knobby enough to get you through the mud if you're careful, but it's still not so knobby that you scrub it down on a 100+ km ride to and from the dirt. In fact, I got about 7000 km out of my last K270 rear, where I was about 80/20 road/dirt. I'm not accelerating hard though, so that helps extend their life. So does proper air pressure.
How do they ride on the road? Well, it takes some getting used to the sideways "walking" feeling of the knobs when you corner hard--especially on pavement. I learned to get used to that by doing hours of figure-8's in a parking lot. Now I feel much more confident on the dirt.
Mountain bike tires are a whole different matter. I ride Kenda Nevegals... the rubber is super grippy on slimy granite and they aren't so deep that they clog up with our regional clay. I've tried tons of tires on my mountain bikes and these are by far the best overall tire for Eastern Ontario so far.
It irks me that my mountain bike tires cost about $60 each, which is more than a rear K270 for my motorcycle. And I get way less life out of a bicycle tire.