DO-AHEAD TIP: Soak bamboo skewers in water for 30 minutes before using, to prevent burning on the grill.
Preheat outdoor or indoor grill to HIGH and lightly oil grill grates.
In a small bowl, combine first 8 ingredients (olive oil through black pepper); blend to a thick paste. Rub about 1/2 tablespoon of mixture evenly over each fillet.
Place remaining paste in a large zipper-topped plastic bag; add mushrooms, red onion quarters and sliced zucchini; gently squeeze the bag to coat the veggies then alternately thread them onto the skewers.
Grill the snapper fillets for 5 minutes then turn and grill for 2 minutes or until fish flakes easily when tested with a fork. Grill the vegetable skewers at the same time as the snapper, cooking for 2 to 3 minutes per side and turning often so as to cook evenly (but not to char).
Serve over spinach and enjoy!