This paper presents the application of wave propagation characterization for complex composite structures through the Algebraic K-space Identification technique in the Cartesian coordinate system (AKSI-C). The proposed method is a novel development since the developed methodology uses adequate partial differential algebraic operations to provide a robust and low-cost framework for identifying wave propagation parameters of the structures with multidimensional signals for the first time. Additionally, the proposed method has been experimentally applied to identify the complex wave propagation phenomenon of different complex composite structures: (i) honeycomb sandwich composite structure: variability of orthotropic behavior and damping properties with frequency and direction is identified by wavenumber space, 3D dispersion curves, and damping loss factor surface. Then, the contribution of individual layer properties on the changes in dynamic behavior is studied by estimating transition frequency; (ii) locally resonant meta-structure: the effect of the local resonator-induced band gap on the wave attenuation is investigated; (iii) periodic rib-stiffened composite plates: the inner resonance phenomena and their remarkable elastic wave manipulation ability are explored by designing 3D-printed resonators and identifying the mixed-resonance-induced band gap. The proposed method has been compared with other inverse methods to assess its reliability under complex conditions.