Sprinkle cucumber with 1 teaspoon salt; cover and let stand 30 minutes. Rinse well.
Bring water to a boil in a medium saucepan; stir in rice and 1 teaspoon salt. Cover, reduce heat, and simmer 20 minutes or until rice is tender and liquid is absorbed. Combine cucumber, rice, olives, and next 5 ingredients in a large bowl.
Combine olive oil, vinegar, and pepper in a small bowl, stirring well with a wire whisk. Pour olive oil mixture over rice mixture; toss gently. Cover and chill. Serve on lettuce leaves, if desired.