In a large bowl, combine olive oil, oregano, garlic powder, onion powder, paprika, salt, pepper, and lemon juice. Add the chicken breasts and coat them well with the marinade. Cover and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat your grill or a grill pan over medium-high heat. Remove the chicken from the marinade and discard the leftover marinade. Grill the chicken for about 6-7 minutes on each side, or until the internal temperature reaches 165°F. Remove from the grill and let it rest for 5 minutes before slicing.
While the chicken is resting, prepare the tzatziki sauce. In a bowl, mix together the Greek yogurt, dill, minced garlic, and diced cucumber. Season with salt and pepper to taste. Set aside.
Slice the grilled chicken into thin strips. Warm the pita breads on the grill for about 1 minute on each side.
To assemble the gyros, place a generous amount of sliced chicken on each pita. Top with tzatziki sauce, diced tomatoes, sliced red onion, and any additional toppings you prefer.
Roll up the pita around the filling and serve immediately.