If using whole breasts, cut each breast in half. Trim any sinews or excess fat off the chicken breasts and discard. Rinse the chicken breasts under cold running water, then drain and blot dry with paper towels. Generously season the breasts on both sides with salt and pepper. Place the breasts in a nonreactive baking dish.
Combine the mango, cucumber, onion, garlic, and chopped mint in a nonreactive mixing bowl and stir to mix. Stir in the lime juice and olive oil and season with salt and pepper to taste; the mango mixture should be highly seasoned. Pour half of the mango mixture over the chicken breasts, turning to coat both sides. Let the chicken marinate for 30 minutes to 1 hour in the refrigerator, covered, turning the breasts once or twice so that they marinate evenly. Refrigerate the remaining mango mixture, covered; you'll use it as the salsa.
Set up the grill for direct grilling and preheat to high.
When ready to cook, drain the marinade from the chicken breasts and discard the marinade. Brush and oil the grill grate. Arrange the chicken breasts on the hot grate, placing them on a diagonal to the bars. Grill the chicken breasts until cooked through, 4 to 6 minutes per side. To test for doneness, poke a breast in the thickest part with your finger; it should feel firm to the touch.
Transfer the grilled chicken breasts to a platter or plates. Spoon the mango salsa over them and garnish with the mint sprigs. Serve at once.