1 Place the tuna steaks in the marinade and turn to coat. 2  Cover, and refrigerate for at least 30 minutes. 3  Preheat grill for high heat. 4  Lightly oil grill grate. 5  Cook the tuna steaks for 5 to 6 minutes, then turn and baste with the marinade.6  Cook for an additional 5 minutes, or to desired doneness. In a large non-reactive dish, mix together the orange juice, soy sauce, olive oil, lemon juice, parsley, garlic, oregano, and pepper. 2  Place the tuna steaks in the marinade and turn to coat. 3  Cover, and refrigerate for at least 30 minutes. 4  Preheat grill for high heat. 5  Lightly oil grill grate.