Non-destructive evaluation of heterogeneous materials is of major interest not only in industrial but also in biomedical fields. In this work, the studied structure is a three-layered one: A laterally heterogeneous anisotropic solid layer is sandwiched between two acoustic fluids. An original method is proposed to solve the wave equation in such a structure without using a multilayered model for the plate. This method is based on an analytical solution, the matricant, explicitly expressed under the Peano series expansion form. This approach is validated for the study of a fluid-loaded anisotropic and homogeneous plane waveguide with two different fluids on each side. Then, original results are given on the propagation of elastic waves in an asymmetrically fluid-loaded waveguide with laterally varying properties. This configuration notably corresponds to the axial transmission technique to the ultrasound characterization of cortical bone in vivo.