  Let $G$ be a finite abelian group. Bridges and Mena characterized the Cayley
graphs of $G$ that have only integer eigenvalues. Here we consider the
$(0,1,-1)$ adjacency matrix of an oriented Cayley graph or of a signed Cayley
graph $X$ on $G$. We give a characterization of when all the eigenvalues of $X$
are integer multiples of $\sqrt{\Delta}$ for some square-free integer $\Delta$.
These are exactly the oriented or signed Cayley graphs on which the continuous
quantum walks are periodic, a necessary condition for walks on such graphs to
admit perfect state transfer. This also has applications in the study of
uniform mixing on oriented Cayley graphs, as the occurrence of local uniform
mixing at vertex $a$ in an oriented graph $X$ implies periodicity of the walk
at $a$. We give examples of oriented Cayley graphs which admit uniform mixing
or multiple state transfer.
