The study of dispersive behavior of guided waves is of great interest in non-destructive testing and development of inspection systems. In this work, a semi-analytical numerical formulation for the computation of dispersion curves and mode shapes in plate-like structures is presented. The formulation is based on a discontinuous Galerkin Finite Element Method. The discretisation of the ordinary differential system in the through-thickness direction yields an eigenvalue problem. The resolution of the latter for a frequency range provides the dispersion curves. The study focuses on Lamb modes in functionally graded material plates with traction-free boundary conditions. The influence of the gradient variations is demonstrated. First, a multilayered approximation consisting of homogeneous laminate is used. Afterwards, the gradients of the parameters are added to the formulation. Numerical examples are presented and compared with those found in the literature. The results are free of spurious modes and show an excellent agreement for all cases, specially for continuously varying properties. The method is very efficient and numerically stable for small and large wave numbers. It was found that a high accuracy is obtained when high-order elements are used on a relatively coarse mesh.