An exact method for evaluating the periods and stability of limit cycles in decentralized relay systems is presented. The method can cope with relays having dead zones and hystereses. The underlying idea is to convert the continuous decentralized relay system under a limit cycle to an equivalent fictitious sampled-data system with synchronous samplers. Consequently, regular sampled-data tools can be applied in a straightforward manner to derive closed-form necessary conditions as well as stability conditions and other properties of such systems. This paper extends and generalises recent results obtained by the authors.