In a saucepan, combine 1 1/3 cups uncooked white rice and 2 2/3 cups water.
Bring the mixture to a boil, then reduce heat to low.
Cover the saucepan and let the rice simmer for 20 minutes.
Meanwhile, place 4 skinless boneless chicken breast halves in a large saucepan.
Fill the pan with water and bring it to a boil.
Cook the chicken for 20 minutes or until done.
Remove the chicken from the water and let it cool.
Once cooled, cut the chicken into bite-sized pieces.
Preheat the oven to 350°F (175°C).
Lightly grease a 9x13 inch baking dish.
In a medium bowl, combine 2 cups shredded Monterey Jack cheese and 2 cups shredded Cheddar cheese.
In a separate bowl, mix together 1 can condensed cream of chicken soup, 1 can condensed cream of mushroom soup, 1 chopped onion, and 1 1/2 cups mild salsa.
Layer half of the cooked rice in the prepared baking dish.
Add half of the cooked chicken pieces on top of the rice.
Spread half of the soup and salsa mixture over the chicken.
Sprinkle half of the cheese mixture on top.
Repeat the layers, ending with the cheese mixture.
Bake the casserole in the preheated oven for about 40 minutes or until bubbly.