Prepare the coals for grilling or light the grill.
In a small bowl, combine the mayonnaise, lemon juice, sesame oil, ginger, mustard, garlic powder, and pepper. Set aside.
Cut each pita bread in half to form 6 pita pockets. Lightly brush the insides of the pita pockets with ginger sauce. Wrap in aluminum foil and place on grill to warm. Remove to outside edge of grill just to keep warm.
Lightly coat the tuna fillets with cooking spray. Pat with pepper and grill on both sides until pink in middle, about 3 minutes per side. Remove from the grill.
To serve, place a fillet in each pocket. Top with lettuce, tomato, and onion slice.