The problem of limit cycles in decentralized relay systems is revisited. An exact method for evaluating the periods of limit cycles and determining their stability in such systems is presented. Though the necessary conditions derived in this paper are equivalent to the existing and well-known infinite series counterpart conditions, the new conditions are constructive and easier to apply since they are given in closed form. The new method is applicable to any system and uses the Z transform familiar to most control engineers and for which extensive tables can be found in most textbooks on discrete time control. Examples demonstrate the usefulness of the new conditions. The new method also reveals an interesting property of decentralized relay systems.