The asymptotic spread properties are investigated for an SIRS reaction-diffusion infectious disease model simulating the disease propagation in temporally periodic environment by using the theory of asymptotic speed of spread. Different from two-dimensional SI type systems, the R component in current model cannot be decoupled from the entire system. This means to overcome the difficulties caused by the coupling of high-dimension and non-autonomy to establish the existence of asymptotic speed of spread for current high-dimensional system. Firstly, the asymptotic spread characteristics of I component is obtained in disease-free region by using the abstract theory of asymptotic spreading speed of monotone systems and comparison principle. Further，the asymptotic spread property of R component in disease-free region is verified through the use of the entire solution and the maximum principle. Secondly, the propagation properties of the invaded region are analyzed by applying comparison principle and uniformly persistent idea to I and R equations, respectively. As a result, the value of asymptotic spreading speed by which we divide the two regions is obtained. In addition, numerical experiments are tested for more specific propagation dynamics of the invaded area in temporally periodic environment.