This paper addresses the application of the novel method of polynomial system resultants for solving two problems governed by systems of cubic equations. Both problems emerge in analysis of stationary dynamics of a periodic waveguide, which consists of linearly elastic continuous rods with nonlinear springs between them. The first one is the classical problem of finding “backbone curves” for free nonlinear vibrations of a symmetric unit periodicity cell of the waveguide. The second one is the problem of finding the Insertion Losses for a semi-infinite waveguide with several periodicity cells. Similarly to the canonical linear case, a very good agreement between boundaries of high attenuation frequency ranges and eigenfrequencies of a unit cell is demonstrated.