Heat grill to medium-high. In a small bowl, combine the cilantro, garlic, lime juice, sugar, and 1 tablespoon oil to make a vinaigrette. Set aside.
Rub the chicken with 1 teaspoon oil and season with ½ teaspoon salt and ¼ teaspoon pepper. Grill until cooked through, 5 to 6 minutes per side.
Meanwhile, in a large bowl, toss the green beans and tomatoes with the remaining 2 teaspoons oil and ¼ teaspoon each salt and pepper.
Grill the beans and tomatoes, turning the beans often and the tomatoes once, until tender and charred, 3 to 5 minutes.
Serve the chicken and vegetables with the vinaigrette.