It is easy for landlords and tenants to end a month-to-month tenancy in Arizona. (The situation is more complicated when it comes to breaking a fixed-term lease.)
In most situations your landlord does not need to give you a reason (although acting on discriminatory or retaliatory motives is illegal). A landlord can simply give you a written notice to move, allowing you 30 days as required by Arizona law and specifying the date on which your tenancy will end.
Your landlord may legally provide less notice in specific circumstances--for example, if you have not paid rent, if you have violated other terms of your rental agreement (for example, bringing in an unauthorized tenant), or if you have violated basic responsibilities imposed by law (such as by dealing drugs on the rental property).
It is equally easy for tenants in Arizona to get out of a month-to-month rental agreement. You must provide the same amount of notice (30 days) as the landlord (unless your rental agreement provides for a shorter amount of notice). Be sure to check your rental agreement which may require that you give notice on the first of the month or on another specific date.
In some situations, you may be able to move out with less (or no) notice—for example, if your landlord seriously violates the rental agreement or fails to fulfill legal responsibilities affecting your health or safety.
Check Arizona state law (Ariz. Rev. Stat. Ann. § 33-1375) for the exact rules and procedures for how landlords must prepare and serve termination notices and for any special rules regarding how tenants must provide notice. See the Laws and Legal Research section of Nolo for advice on finding and reading statutes and court decisions.