Marinate the Chicken: Whisk together olive oil, minced garlic, lemon juice, dried oregano, smoked paprika, salt, and pepper in a bowl. Add chicken strips and coat evenly. Cover and marinate in the refrigerator for at least 30 minutes, preferably overnight.
Grill the Chicken: Preheat your grill or grill pan over medium-high heat. Remove marinated chicken from the refrigerator and grill for 5-7 minutes on each side until fully cooked (internal temperature should reach 165°F).
Prepare Tzatziki Sauce: Combine grated cucumber (squeezed of excess moisture), yogurt, minced garlic, lemon juice, dill, salt, and pepper in a bowl. Mix well and chill until ready to use.
Warm Pita Bread & Assemble Gyros: Warm pita bread on the grill or in a microwave until soft. Layer grilled chicken slices, tzatziki sauce, and fresh veggies on each pita.
Roll & Serve: Roll up each pita tightly around its filling and serve immediately while warm.