We have found exact, periodic, time-dependent solitary wave solutions of a discrete phi4 field theory model. For finite lattices, depending on whether one is considering a repulsive or attractive case, the solutions are Jacobi elliptic functions, either sn (x,m) [which reduce to the kink function tanh (x) for m-->1 ], or they are dn (x,m) and cn (x,m) [which reduce to the pulse function sech (x) for m-->1 ]. We have studied the stability of these solutions numerically, and we find that our solutions are linearly stable in most cases. We show that this model is a Hamiltonian system, and that the effective Peierls-Nabarro barrier due to discreteness is zero not only for the two localized modes but even for all three periodic solutions. We also present results of numerical simulations of scattering of kink-antikink and pulse-antipulse solitary wave solutions.