In a large bowl, whisk together olive oil, red wine vinegar, lemon juice, garlic, oregano, salt, black pepper, and paprika. Add the chicken thighs and toss to coat. Cover and marinate in the refrigerator for at least 1 hour, or up to overnight for best flavor.
Preheat your grill or a grill pan over medium-high heat. Remove the chicken from the marinade and grill for about 6-7 minutes per side, or until the chicken is cooked through and has nice grill marks. The internal temperature should reach 165°F.
While the chicken is grilling, prepare the tzatziki sauce by combining Greek yogurt, diced cucumber, red onion, parsley, dill (if using), and salt and pepper in a bowl. Mix well and set aside.
Once the chicken is cooked, let it rest for a few minutes before slicing it into thin strips.
To assemble the gyros, warm the pita breads on the grill for about 1 minute on each side. Place a generous amount of sliced chicken in the center of each pita, top with tzatziki sauce, and add additional cucumber, red onion, and parsley if desired.
Roll up the pita around the filling and serve immediately.