I don't think they all stink. I recently bought a 17" Olfa rotary mat (great mat, by the way!) and it had no smell out of the box. It's probably a good idea to purchase, in-person, only mats that are exposed to air so they can be given the "sniff test" before purchasing.
There are some plastics that outgas so much I won't have them in the house! Outgassing is a chemical process that can take a year to complete. Putting the mat in the sun will probably only increase the chance of it warping. It may need to be left outside to air for a year before it stops outgassing. This is similar to new-car and new-house outgassing. Google outgassing or offgassing of plastics for more info.