e have developed a theory for experimentally found resonances in one-dimensional discrete Josephson transmission lines. These resonances appear when both the inductive and capacitive interactions between junctions occur. The spectrum of linear electromagnetic waves propagating in this system contains many bands. We calculate the resonance voltages by making use of a mapping to the theory of Brillouin bands. The dependence of the resonance voltages on the external magnetic field is found to be in good accord with experiments on strip-line Josephson junction arrays and long Josephson junctions with periodically distributed microresistors.