Influence of the linear energy-momentum relationship in graphene on conductance and magnetoresistance (MR) in ferromagnetic metal (FM)/graphene/FM lateral junctions is studied in a numerical simulation formulated using the Kubo formula and recursive Green’s function method in a tight-binding model. It is shown that the contribution of electron tunneling through graphene should be considered in the electronic transport in metal/graphene/metal junctions, and that the Dirac point (DP) is effectively shifted by the band mixing between graphene and metal electrodes. It is shown that MR appears due to spin-dependent shift of DP or spin-dependent change in the electronic states at DPs. It is shown that the MR ratio caused by the latter mechanism can be very high when certain transition metal alloys are used for electrodes. These results do not essentially depend on the shape of the junction structure. However, to obtain high MR ratios, the effects of roughness should be small.