Plane time-harmonic elastic SH-wave propagation through an isotropic inhomogeneous layer surrounded by two homogeneous half-spaces is studied in this article. The material properties of the inhomogeneous layer are assumed to be non-uniform along the thickness direction according to a distribution law described by the triconfluent Heun functions or their polynomial forms that contain a number of optional parameters. The general analytical solution of the governing equation for elastic SH-waves in the layer is presented. Employing optional parameters, the material-property profiles can be varied to a relatively large extent without the need to seek new solutions of the governing equation for a chosen material-property profile. If the wave speed is constant in the inhomogeneous layer, the derived analytical solution is exact; otherwise the analytical solution is approximate. As a part of this article, the method enabling to find an approximate analytical solution of the governing equation for predetermined material functions is also presented. The applicability of the analytical solutions are tested and discussed based on the representative examples, and at the same time, the analytical results are compared with numerical ones to demonstrate their validity.