I've toured on a mountain bike and I've toured on a touring bike. I enjoy both, but for me, the touring bike wins out. Mine is able to handle everything from good pavement to packed dirt trails comfortably, which is important considering some of the places I've gone. I have campus pedals and I use shoes with recessed cleats. I wouldn't want to jog in my cycling shoes, but I can walk long distances in them. A good pair of shoes will be comfortable off the bike as well as on the bike.

The point is that touring by bike is incredibly flexible. If a bike is roadworthy, there is probably someone touring on it or on something similar. There is no definitive right way to do this.