Preheat grill for medium heat and lightly oil the grate.
Combine olive oil, 2 tablespoons lemon juice, water, 2 teaspoons Italian seasoning, 1 teaspoon lemon zest, oregano, salt, and pepper in a bowl.
Place cucumber, green pepper, red onion, artichoke hearts, and garlic in a large bowl. Toss salad to distribute evenly. Cut each pita bread round into 4 to 5 slices.
Toast pita bread on the preheated grill until slight grill marks appear, about 4 minutes.
Place chicken in a grill pan and coat with the remaining 1 tablespoon lemon juice, 1 tablespoon lemon zest, salt, and pepper. Transfer the pan to the grill.
Grill the chicken until no longer pink in the center and the juices run clear, 5 to 10 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Arrange chicken, salad, and pita bread on a serving plate and enjoy sandwich-style or as individual components.