Combine olive oil, oregano, basil, lemon juice, garlic, salt, pepper, and chicken cubes in a container. Marinate for at least 2 hours.
While chicken is marinating, soak rosemary in water for the same amount of time.
After two hours, slice zucchini and summer squash (about 1in ) and set aside. Open artichokes and drain. Cut each rosemary stem so that end comes to a point. Add meat and vegetables alternately. Be sure to leave about three inches one each end. Place finished skewers on a plate until ready to cook.
Heat grill to medium-high (if using charcoal, be sure coals are hot enough). Place skewers near edge of the grill and all end of rosemary to be outside the cover. Cover (if using grill without a lid, a piece of foil will work well) and cook 5-10 min on each side. Monitor to prevent burning.
Remove from grill when chicken is no longer pink in center.
Line oven rack with foil and curl edges to prevent dripping. Set oven to Broil. Place skewers on rack and cover with another piece of foil. Cook 5-10 minutes. Monitor to prevent burning.
Remove from oven when chicken is no longer pink in center.
Serve this dish with rice for a great summer meal!