It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. A particularly interesting necessary condition for the existence of a digraph of degree three and diameter $k\ge 3$ of order one less than the Moore bound is that the number of its arcs be divisible by $k+1$. In this paper we derive a new necessary condition (in terms of cycles of the so-called repeat permutation) for the existence of such digraphs of degree three. As a consequence we obtain that a digraph of degree three and diameter $k\ge 3$ which misses the Moore bound by one cannot be a Cayley digraph of an Abelian group.