Preheat oven to 350°F. Stack tortillas and wrap in foil. Bake about 10 minutes or until softened. Meanwhile, in a medium bowl, combine mozzarella cheese, provolone cheese, olives, nuts, and the 2 teaspoons snipped oregano or ½ teaspoon dried oregano. Spread cheese mixture onto half of each tortilla. Fold tortillas in half; secure with wooden toothpicks. Brush one side of each quesadilla with some of the oil.
In a large skillet or on a griddle, place quesadillas, 2 or 3 at a time, oiled sides down; cook over medium heat about 4 minutes or until heated through, brushing with remaining oil and turning once. Cut each quesadilla in half.
Meanwhile, in a small saucepan, heat salsa just until hot; stir in the 1 teaspoon snipped oregano or ¼ teaspoon dried oregano. If desired, garnish with oregano sprigs. Serve with quesadillas.