Regular representations of finite groups, as introduced by Cayley, are among the most natural permutation representations of finite groups. Thus, the question which regular representations appear as full automorphism groups of combinatorial structures has been addressed and resolved for several classes of structures, notably for graphs (where they are called Graphical Regular Representations, GRR's), digraphs (Digraphical Regular Representations, DRR's) as well as for hypergraphs allowing for hyperedges of varying sizes. In the present paper, we focus on $k$-hypergraphs, which are hypergraphs in which all hyperedges are of the same size $k$, and address the question which $k$-regular hypergraphs possess full automorphism groups acting regularly on the vertices. We rely on the concept of a Cayley hypergraph (defined here) and show that all sufficiently large finite groups admit a regular representation as the full automorphism group of a $3$-hypergraph.