Heat a grill pan over medium heat or preheat a charcoal or gas grill to medium. (If you're using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 to 6 seconds.) Toss tuna with 1 tablespoon oil. Thread tuna onto 4 skewers.
Combine cilantro, lime juice, and 1/4 cup oil. Season with salt and pepper. Reserve 3 tablespoons.
Brush pan or grill grate with remaining tablespoon oil. Grill tuna, turning skewers and brushing tuna occasionally with lime sauce, to desired doneness, about 2 minutes per side for rare. Transfer skewers to a serving plate, and brush with reserved lime sauce. Serve immediately.
If you're using wooden or bamboo skewers, soak them in cold water for at least 30 minutes before using.