Make the marinade. In a mixing bowl, combine olive oil, lemon juice, minced garlic, white pepper, Greek seasoning, kosher salt and black pepper. Mix well until all the ingredients are thoroughly combined.
Marinate the chicken. Place the chicken cubes in a large resealable plastic bag or a shallow dish. Pour the marinade over the chicken, ensuring all pieces are evenly coated. Seal the bag or cover the dish with plastic wrap. Marinate in the refrigerator for at least 1 hour, preferably overnight, to allow the flavors to penetrate the chicken.
Prep the skewers. If using wooden skewers, soak them in water for about 30 minutes to prevent them from burning during cooking.
Prep the grill. Preheat your grill or grill pan over medium-high heat.
Prep the chicken. Thread the marinated chicken pieces onto the skewers, alternating with chopped onion pieces.
Grill. Place the skewers on the preheated grill or grill pan. Cook for 6-8 minutes per side, or until the chicken is cooked and has grill marks. Rotate the skewers occasionally for even cooking.
Rest. Once cooked, remove the skewers from the grill and let them rest for a few minutes.
Serve. Serve the chicken souvlaki hot with your favorite sides, such as pita bread, tzatziki sauce, or Greek salad. Garnish with fresh herbs such as fresh mint, fresh oregano, or fresh dill.