Cut the chicken breasts into bite-sized 1-inch cubes and place them in a large bowl.
In a measuring cup, whisk together the extra virgin olive oil, lemon juice, dried oregano, minced garlic, salt, and black pepper.
Pour the marinade over the cubed chicken. Toss the chicken pieces until they are evenly coated.
Cover the bowl and refrigerate for at least 2 hours, or up to overnight for maximum flavor.
If using bamboo skewers, soak them in cold water for 30 minutes before threading the chicken.
Remove the chicken from the marinade and thread 5 to 6 pieces loosely onto each skewer, discarding any excess marinade.
Preheat the grill to medium-high heat. Lightly oil the grates.
Grill the chicken skewers for 10 to 15 minutes, turning occasionally, until the chicken is cooked through and lightly charred.
Serve hot with tzatziki sauce, cucumber slices, fresh dill, and warm pita bread.