Marinate chicken: In a large bowl or zip-top bag, combine olive oil, lemon juice, minced garlic, red wine vinegar, oregano, paprika, cumin, cinnamon, salt, and pepper. Add chicken strips and toss to coat thoroughly. Cover and refrigerate for at least 2 hours or up to overnight for maximum flavor.

Make tzatziki sauce: Grate cucumber and squeeze out excess moisture using paper towels or cheesecloth - this is crucial for thick sauce. In a bowl, combine Greek yogurt, squeezed cucumber, minced garlic, lemon juice, dill, olive oil, and salt. Mix well. Refrigerate until ready to serve. Sauce tastes better after sitting for 30 minutes.

Cook chicken: Heat large skillet or grill pan over medium-high heat. Remove chicken from marinade, letting excess drip off. Cook chicken strips for 6-8 minutes, turning occasionally, until golden brown and cooked through (internal temperature 165°F). Don't overcrowd the pan - work in batches if needed.

Warm pita bread: Wrap pitas in damp paper towel and microwave for 20-30 seconds, or warm in dry skillet for 30 seconds per side until soft and pliable.

Assemble gyros: Lay warm pita flat. Spread generous amount of tzatziki sauce down the center. Layer with shredded lettuce, diced tomatoes, sliced red onion, diced cucumber, and cooked chicken strips. Sprinkle with crumbled feta cheese.

Wrap and serve: Fold in sides of pita and roll tightly from bottom up. Cut in half diagonally if desired. Serve immediately with extra tzatziki sauce and lemon wedges on the side.