A transmission-line model is developed for the analysis of planar waveguides including wire-medium slabs with vertically aligned wires. The developed network formalism allows for an effective determination of the modal dispersion equation via transverse resonance, as opposed to the more cumbersome fieldmatching approach. Closed and open parallel-plate waveguides loaded with wire medium are considered. The suppression of undesired TEM-mode propagation and the possible achievement of a unimodal regime are investigated. Numerical results, validated also against state-of-the-art simulation software, confirm the accuracy and effectiveness of the proposed approach. Dispersion relations of complex modes are obtained for a class of leaky-wave antennas based on open wire-medium layered structures covered by partially reflecting screens.