Preheat a charcoal or gas grill for indirect grilling over medium heat and oil the grill rack.
Toss chicken with oil, lemon zest, juice, marjoram, & pepper. Marinate in refrigerator 1 to 6 hours.
Stir together onion, garlic, vinegar, salt & pepper, in small bowl. Whisk in olive oil. Set aside.
Remove chicken from marinade, pat dry. Grill chicken strips over medium heat. Turn strips often, cook until no longer pink, or until chicken is firm to touch. Transfer strips onto platter.
Stir fresh mint into the onion sauce, and season to taste. Pour over the grilled chicken strips.