Seems like everyone is pointing to Surly LHT as THE bike to have (ok a few have mentioned Trek 520). Is anyone here willing to admit to using something else? Before starting my research, I assumed I could take my Cannondale fat tube unsuspended bike, add some racks and go, but I'm guessing there are good reasons why not.
"Admit" is a bit strong, as I've got a Novara Randonee and Fuji Touring. Really, most of the mass-produced touring bikes are similar, except that a fair few have higher (than I'd like) gearing.
Lots of people have ridden lots of different kinds of bikes, and there are only a few things that make a bike unsuitable for touring. Weak frames (sold as "lightweight", such as racing bikes and carbon fiber) have a hard time handling the load a loaded tourist put on them. Skinny tires aren't the best with a load, or on dirt, gravel, mud, etc. Suspension frames waste a lot of pedaling energy, unless you can lock the suspension out. Knobs give you a buzz on the road, which is a drag and pretty unpleasant after some distance.
Your unsuspended mountain bike will probably be great for carrying a load on roads. It'd be a good idea to put slick tires on it. You might come to wish for multiple hand positions that road (drop) bars give, or you might never notice. And MTBs are often sold without the attention to fit a good bike dealer will give you when you buy a road bike; sub-optimal fit might limit your daily time in the saddle.
That said, get some racks and change out your tires, then try some weekend and week-long trips. You'll soon figure out if you need to make any other changes.