Combine mushroom caps and teriyaki sauce in a heavy-duty zip-top plastic bag, turning to coat; seal and let stand 20 minutes.
Grill mushroom caps, covered with grill lid, over medium-high heat (350° to 400°) 2 minutes on each side. Top with cheese, and grill 2 minutes.
Spread mayonnaise on cut sides of buns. Grill buns, cut sides down, 1 minute or until toasted. Place mushrooms in buns, and serve immediately.