An oriented triple system of order \(v\), denoted OTS\((v)\), is said to be \(d\)-cyclic if it admits an automorphism consisting of a single cycle of length \(d\) and \(v-d\) fixed points, \(d\geq 2\). In this paper, we give necessary and sufficient conditions for the existence of \(d\)-cyclic OTS\((v)\). We solve the analogous problem for directed triple systems.