Whisk together the lemon juice, olive oil, oregano, salt, and pepper in a large zip-top plastic bag. Add the garlic, onion, and chicken, zip the bag closed, and squish everything around so the chicken is coated. Put the plastic bag inside a bowl and refrigerate 30 minutes, or up to 2 hours.
Thread the chicken and onion onto skewers and grill (on an outdoor grill or an indoor grill pan) until fully cooked, about 8 minutes, flipping once halfway through.
To make a Greek chicken kebab platter, serve the chicken on a bed of lettuce along with fresh herbs, tomato, olives, feta cheese, tzatziki sauce, and/or pita bread.