By solving the two-component spinor equation for massless Dirac fermions, we show that graphene under a periodic external magnetic field exhibits a unique energy spectrum. At low energies, Dirac fermions are localized inside the magnetic region with discrete Landau energy levels, while at higher energies, Dirac fermions are mainly found in non-magnetic regions with continuous energy bands originating from wavefunctions analogous to particle-in-box states of electrons. These findings offer a new methodology for the control and tuning of massless Dirac fermions in graphene.