Let S be an orthogroup, that is a completely regular semigroup the idempotent set of which is a subsemigroup. By a result of \textit{M. Petrich} [Proc. Am. Math. Soc. 99, 617-622 (1987; Zbl 0622.20050)], S is a semilattice Y of semigroups \(S_{\alpha}=I_{\alpha}\times G_{\alpha}\times \Lambda_{\alpha}\), \(\alpha\in Y\), where \(I_{\alpha}\), \(\Lambda_{\alpha}\) are left zero and right zero semigroups respectively, and \(G_{\alpha}\) are groups. Let R be a ring and \(\pi\) a hereditary supernilpotent radical. The radical \(\pi\) (R[S]) of the semigroup ring R[S] is described in terms of certain congruences on S and of \(\pi\) (R[G]) where G is the induced semilattice of groups \(G_{\alpha}\), \(\alpha\in Y\). To complete the description of \(\pi\) (R[S]) one can thus use a characterization of \(\pi\) (R[G]) obtained by \textit{I. S. Ponizovskij} [in Semigroup Forum 28, 143-154 (1984; Zbl 0527.20049)]. As a consequence, it is shown that the Jacobson radical of R[S] is the sum of the Jacobson radicals of all \(R[S_{\alpha}]\), \(\alpha\in Y\), provided that all \(G_{\alpha}\) are periodic groups.