Following recent experimental studies, we present in this paper a linear model of the periodic-waveguide cyclotron maser. This device is based on a cyclotron-type interaction of a nonrelativistic electron beam with traveling waves in a metallic periodic waveguide. The kinetic model presented in this paper incorporates the periodic-waveguide spatial wave harmonics with the cyclotron and Weibel interactions. It results in a Pierce-type gain-dispersion equation with new coupling terms. The dependence of the cyclotron amplification on the low impedance of the inductive periodic waveguide and on the electron initial transverse velocity is analyzed in various operating regimes. In agreement with experimental results, this analysis shows that amplification is feasible in certain conditions without an initial electron transverse velocity (V0=0) and with a wide acceptance of electron beam energy variation.