In a bowl, combine garlic powder, onion powder, oregano, paprika, salt, pepper, olive oil, and lemon juice. Mix well to create a marinade. Add chicken breasts and coat thoroughly. Cover and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat a grill or grill pan over medium-high heat. Remove chicken from marinade and discard the marinade. Grill chicken for 6-7 minutes per side, or until cooked through and internal temperature reaches 165°F. Remove from heat and let rest for 5 minutes.
While the chicken is resting, prepare the yogurt sauce by mixing Greek yogurt, dill, and lemon juice in a small bowl. Season with salt to taste.
Slice the grilled chicken into thin strips.
To assemble the gyros, warm the pita breads on the grill for about 1 minute per side. Place sliced chicken in the center of each pita, then top with diced cucumber, cherry tomatoes, red onion, and a generous dollop of yogurt sauce.
Roll the pita around the filling and serve immediately.