In this paper an effective quasi-analytical field theory method for the analysis ofmultisection rectangular waveguide structures with E or H plane step discontinuities is presented. A version of Galerkin's method is used to solve the corresponding diffraction problems. The weghted Gegenbauer polynomials are used as basis functions, taking into account the field asymptotic at the discontinuity edges. Another important feature of the solution is the separation and analytical transform of the matrix operator static part of the obtained linear algebraic equations set. The method proposed here guarantees rapid convergence and high accuracy of the numerical results. It has been verified by numerical convergence investigation and comparison with the results of other works. The method can be applied for the effective design of a wide range of passive waveguide components: different band-pass, stop-band, and low-pass filters, waveguide phase shifters, polarizers, etc. As examples of the concrete devise design, the characteristics of the inductive and capacitive iris filters, evanescent-mode waveguide dielectric resonator filter, and E plane stub-loaded, fixed-phase shifter are given in the paper.