In a 3- to 4-quart pan, combine wheat berries, rice, and 6 cups water. Bring to a boil over high heat. Cover and simmer until grains are tender to bite, 30 to 35 minutes. Drain. Pour into a large bowl.
Drain artichoke marinade into bowl with grains. Coarsely chop artichokes and add to bowl along with vinegar, oil, mustard, and oregano. Mix well and let stand until cool, about 20 minutes.
Add tomatoes, cucumber, onions, cheese, olives, mint, and parsley; mix well. Add salt and pepper to taste.