DO-AHEAD TIP: Marinate snapper fillets for 30 minutes (see recipe).
Preheat grill to MEDIUM.
Place fish in a shallow dish. In a small bowl, whisk together remaining ingredients except rice. Rub mixture into fillets then cover and marinate at room temperature for 30 minutes.
Meanwhile, prepare brown rice. Lightly oil the grill grate. Grill fillets over indirect heat for 5 to 8 minutes, turning once, until fish flakes easily when tested with a fork.
Watch carefully to prevent burning! Serve and enjoy!