In this work we theoretically study microwave properties of a superconductor/dielectric periodic layered structure, in which a strongly dispersive superconductor, nearly ferroelectric superconductor (NFE SC), is taken. Microwave transmittance in the dielectric-like response has been calculated based on the transfer matrix method as well as the electrodynamics of NFE SCs. Microwave response is strongly dependent on the number of periods as well as the thickness of superconductor layer. It is found that the first anomalous transmission peak can be created when the number of periods is more than five. In addition, more anomalous peaks are generated by greatly increasing the number of periods. The presence of anomalous sharp peaks can be used to design a nicely frequency-selective filter or sampler using such a multilayer structure.