Cook brown rice according to package instructions, about 30-40 minutes. Let cool slightly.
Cook chicken breasts thoroughly in a skillet or bake at 375°F for 20-25 minutes. Chop into small pieces. Alternatively, cook ground turkey until browned.
Heat refried beans in a skillet over medium heat until warm for easier spreading.
In a large mixing bowl, combine cooked chicken, warmed refried beans, brown rice, salsa, and chopped cilantro. Stir in half the Greek yogurt to make a creamy filling.
Preheat oven to 350°F. In a baking dish, layer whole-wheat tortillas, half the filling, and shredded cheddar cheese. Repeat layers with remaining tortillas and filling, finishing with a layer of tortillas.
Spread remaining Greek yogurt over top layer of tortillas and sprinkle with remaining cheddar cheese.
Cover dish with foil and bake for 25 minutes. Remove foil and bake for another 10-15 minutes until cheese is bubbly and golden.
Let casserole cool a few minutes. Slice into squares, garnish with fresh cilantro, and serve with extra salsa or Greek yogurt if desired.