We endeavored to investigate directed strongly regular Cayley graphs (or DSRCG for short) over dicyclic groups ${\mathrm{Dic}}_{4n}=⟨\alpha ,\beta |{\alpha }^n={\beta }^4=1,{\beta }^{-1}\alpha \beta ={\alpha }^{-1}⟩$, where $n$ is odd. We derived several DSRCGs over ${\mathrm{Dic}}_{4n}$ for $n$ odd. We then derived a criterion for a certain class of Cayley graph to be directed strongly regular.