Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2
\rfloor}, and let I(G) denote its the edge ideal in the ring R =
k[x_1,...,x_n]. We consider the problem of determining when G is
Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay
graph G must be well-covered, we focus on known families of well-covered
circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic
circulant graphs are Cohen-Macaulay. We end with the observation that even
though the well-covered property is preserved under lexicographical products of
graphs, this is not true of the Cohen-Macaulay property.