In a medium bowl, whisk together the olive oil, lemon juice, minced garlic, oregano, salt, pepper, and red wine vinegar until combined.
Add the chicken cubes to the bowl and toss until every piece is coated. Cover and refrigerate for at least 30 minutes or up to 4 hours. Longer than 4 hours can make the texture soft because of the acid.
If using wooden skewers, soak them in water for 20–30 minutes so they do not burn. Thread the marinated chicken onto the skewers, leaving a little space between pieces. Alternate with vegetables if using.
For outdoor grilling, heat the grill to medium-high (about 400–450°F). For the oven, set the broiler to high and place a rack about 6 inches from the heat. You can also use a stovetop grill pan over medium-high heat.
Grill or broil the skewers for 10–15 minutes, turning every 3–4 minutes. Look for light char marks and golden edges. The chicken is done when it reaches 165°F in the center and the juices run clear.
Let the skewers rest for 3 minutes. Squeeze fresh lemon over the top and sprinkle with parsley or dill. Serve with tzatziki on the side.