In a food processor; pulse the basil, garlic, salt and pepper. Stream in the olive oil until it comes together. Marinate the chicken pieces in the pesto for at least 30 minutes.
Preheat grill to high heat.
Skewer marinated chicken and tomatoes alternating between the two until you use up all the ingredients.
Grill skewers about 4 minutes on each side. Remove from grill and serve.