It is shown that in organic molecular crystals the fluctuations of the intermolecular electron jump frequencies resulting from the lattice vibrations may produce a considerable increase of the total electric current. An expression for the current density operator, which takes into account the electron-phonon interactions, is derived and the corresponding linear response theory of the electrical conductivity is formulated. The phonon-assisted current can be partially accounted for by the proper corrections of the electron velocities used in a simple band theory calculation. These corrections are explicitly calculated for a three-dimensional parabolic and a one-dimensional narrow electron energy band.