An analytical study on transverse surface waves propagating in a functionally graded piezoelectric substrate carrying a metal layer of finite thickness was carried out. Dispersion relations for the existence of the waves were obtained and the effects of material gradient on wave propagation were quantified. Numerical examples show that the presence of the material gradient affects significantly the fundamental mode but has only negligible effects on the higher order modes. Depending on whether the surface wave velocity is smaller or greater than the bulk shear wave velocity in the metal layer, three different types of the dispersion behavior are discussed.