  Higher-rank graphs were introduced by Kumjian and Pask to provide models for
higher-rank Cuntz-Krieger algebras. In a previous paper, we constructed
2-graphs whose path spaces are rank-two subshifts of finite type, and showed
that this construction yields aperiodic 2-graphs whose $C^*$-algebras are
simple and are not ordinary graph algebras. Here we show that the construction
also gives a family of periodic 2-graphs which we call \emph{domino graphs}. We
investigate the combinatorial structure of domino graphs, finding interesting
points of contact with the existing combinatorial literature, and prove a
structure theorem for the $C^*$-algebras of domino graphs.
