The dynamics of the electrons that propagate in graphene superlattices is described by a pseudospinor that satisfies a massless Dirac-type equation, which, in general, can be very difficult to solve. Using effective medium theory techniques, it was recently shown that under certain conditions the microscopic Hamiltonian of the system can be approximated by an effective Hamiltonian. In this paper we develop a finite difference in the time domain (FDTD) numerical scheme to solve the pertinent time evolution problem, and compare the results obtained using both microscopic and macroscopic approaches. It is shown that despite the simplicity of the effective medium method, it still provides accurate physical results.