Rinse steaks under cold water and pat dry. Place in zip-top bag and add red wine, dressing mix, and pepper. Squeeze air out of bag and seal. Gently massage bag to combine ingredients. Marinate in refrigerator for 1 to 3 hours
Remove steaks from marinade and discard marinade. With a sharp knife cut a pocket into the side of each steak, being careful not to cut through. Stuff steaks with mushroom mixture and secure with toothpicks. Place on hot, oiled grill and cook 5 to 7 minutes per side for medium (160 degrees F). Remove from grill.
To serve, remove toothpicks. Garnish steaks with sprigs of fresh oregano and serve hot.