This paper presents some useful discussions on the shear horizontal acoustic waves in an inhomogeneous plate that possess coupled piezoelectric, piezomagnetic, and magnetoelectric effects. The magnetoelectroelastic material is of hexagonal symmetry (6mm crystal) and polarized in the shear horizontal direction. It is assumed that the material constants of the magnetoelectric plate vary continuously along the thickness direction. Solutions of the mechanical displacement, electric potential, and magnetic potential are obtained by solving the coupled field equations. The phase velocity equations are obtained and the influence of the inhomogeneity of the material constants on the phase velocity is considered. These findings are significant in the applications of wave propagation in the coupled piezoelectric, piezomagnetic, and magnetoelectric structures.