Simulations of exact spin dynamics in systems of multiple nuclear spins are limited to relatively small numbers of spins by the large size of the Hamiltonian matrices for dipolar-coupled spin systems. We propose using the translational symmetry of periodic (crystalline) systems to block-diagonalize both the Hamiltonian and the density matrix and thus extend the range of exact calculation. Although this method has general application whenever the density matrix treatment is valid, the method is illustrated using simulations of spin diffusion in one-dimensional spin-1/2 lattices. Clear evidence of convergence towards the limit of an infinitely extended lattice is seen.