There are some good classification of Integral Cayley graphs, which by your terminology means all their adjacency matrix eigenvalues are integer. The adjacency matrices of Cayley graphs over cyclic groups is circulant.
Prof. Alireza Abdollahi and Dr. Vatandoost did some good classification of such graphs in the paper with name:
"Which Cayley Graphs are Integral?, The electronic journal of combinatorics 16 (2009), #R122"
Also, there is an other good resource for study the classification of integral Cayley graphs, which is:
"Integral Cayley graphs and groups, A. Ahmady, J. P. Bell, B. Mohar"
Also, you can see the paper:
"On groups all of whose undirected Cayley graphs of bounded valency are integral"
There are some good references in these three papers that I mentioned above.