The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions, and has applications to representation theory. In this paper, we enumerate domino tableaux of shape 2-by-n, and use this result to prove a new CSP on these objects. We then enumerate the rectangular domino tableaux of any dimensions, and conjecture a more general CSP on rectangular domino tableaux. As a consequence of the enumerative results, we obtain several identities involving Fibonacci and Catalan numbers.