There are three heavy hitters in Northern Arizona in the cockroach world.
I would say the most common around homes is the oriental cockroach or also known as the water bug. This Guy is not a structure infester. He generally likes to stay out side around the landscaping. At night time they will wander in. This roach is easily controlled with regular pest control service.
The second most common is the American Cockroach. He cohabitates a lot with the oriental cockroach. This Roach can fly and is also easily controlled with regular pest control service.
The third roach is the German cockroach and this is the little guy everybody hates. This roach likes to establish itself in the house around kitchens and bathrooms. Generally you will notice this roach walking across the counter at night when you go to the kitchen for that midnight snack. Very difficult to control and requires homeowner and exterminator working together.