The OPERA collaboration recently reported that the muon neutrinos could be superluminal. More recently, Cohen and Glashow pointed that such superluminal neutrinos would be suppressed since they lose their energies rapidly via bremsstrahlung. Here, we propose that the Finslerian nature of the space time could account for the superluminal phenomena of particles. The Finsler spacetime permits the existence of the superluminal behavior of particles while the causality still holds. A new dispersion relation is obtained in a class of Finsler spacetimes. It is shown that the superluminal speed is linearly dependent on the energy per unit mass of the particle. We find that such a superluminal speed formula is consistent with the data of the OPERA, MINOS and Fermilab-1979 neutrino experiments as well as the observations on the neutrinos from the SN1987A.