Preheat the grill or grill pan over medium-high heat.
Rub the chicken breasts with olive oil, and season with salt and pepper on both sides.
Grill the chicken for about 6-8 minutes on each side, or until cooked through and juices run clear. Remove from heat and let rest for a few minutes.
While the chicken is resting, prepare the tzatziki dressing in a mixing bowl by combining Greek yogurt, minced garlic, lemon juice, dill, parsley, and a pinch of salt. Whisk until smooth.
Cut the grilled chicken into strips or cubes.
In a large mixing bowl, combine the diced cucumber, cherry tomatoes, red bell pepper, and red onion.
Add the chicken pieces to the bowl with vegetables and pour the tzatziki dressing over the top. Toss gently until everything is well coated.
Serve on a platter and garnish with additional herbs if desired.