In medium bowl, combine mango nectar, 6 tablespoons of the olive oil, 4 tablespoons of the lime juice, garlic, sugar and 1/2 teaspoon of the salt. Place chicken breasts and mango mixture in large resealable plastic bag; refrigerate 2 hours.
Peel and dice mango, cucumber and avocado. Finely chop jalapeno. In large bowl, combine mango, cucumber, avocado, jalapeno, cilantro, remaining 2 tablespoons oil, remaining 2 tablespoons lime juice, and remaining 1/2 teaspoon salt. Refrigerate until serving.
Heat gas grill to high, prepare outdoor grill with hot coals, or heat broiler. Remove chicken from marinade. Grill 6 minutes per side or until internal temperature registers 170°F on an instant-read thermometer. Serve with salsa.