The cloth is glued to a fiberboard or masonite frame, which in turn is glued to the cabinet. If they are earlier models, there may be stables into the cabinet thru the cloth and frame. But if they are an early version, I would expect them to have cloth surrounds, rather than foam.
Because I deal with this kind of thing often, I bent the last 3/4 inch of a putty knife almost 90 degrees and use it to work my way around the edges of the grill frame to separate it. Occasionally, I break the frame...easy enough to make new ones, but then of course you also have to get new grill cloth.
As for the staples....if you are careful, and somewhat lucky, you can pull them out without damaging the cloth. If stapled, there will be a lot of them!
I would use velcro tabs to re-install the grills.