We study the electronic states of graphene in piecewise constant potentials using the continuum Dirac equation appropriate at low energies and a transfer matrix method. For superlattice potentials, we identify patterns of induced Dirac points that are present throughout the band structure and verify for the special case of a particle-hole symmetric potential their presence at zero energy. We also consider the cases of a single trench and a p-n junction embedded in neutral graphene, which are shown to support confined states. An analysis of conductance across these structures demonstrates that these confined states create quantum interference effects, which evidence their presence. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.