  A graphical regular representation (GRR) of a group $G$ is a Cayley graph of
$G$ whose full automorphism group is equal to the right regular permutation
representation of $G$. Towards a proof of the conjecture that only finitely
many finite simple groups have no cubic GRR, this paper shows that
$\mathrm{PSU}_3(q)$ has a cubic GRR if and only if $q\geq4$. Moreover, a cubic
GRR of $\mathrm{PSU}_3(q)$ is constructed for each of these $q$.
