The notion of pi-regularity is a generalization of von Neumann regularity. In this paper we begin our investigation of pi-regular and strongly pi-regular semigroup graded rings. We first consider the pi-regularity of those semigroup graded rings whose prime ideals and radicals were characterized by authors such as Bell, Stalder and Teply in recent papers. The classes of semigroups underlying these semigroup graded rings include, as examples, regular bands, Clifford semigroups, commutative power stationary semigroups and more generally commutative semigroups for which every element has a power lying in a subgroup. We then extend our study to Munn rings and inverse semigroup rings. In the case of strong regular band graded rings we characterize pi-regularity of the ring in terms of pi-regularity of its homogeneous components. For rings graded strongly by Clifford semigroups as well as for Munn rings and inverse semigroup rings, the study of pi-regularity is reduced to the case of group rings.