The dynamic propagation of a crack in a functionally graded piezoelectric material (FGPM) interface layer between two dissimilar piezoelectric layers under anti-plane shear is analyzed using integral transform approaches. The properties of the FGPM layers vary continuously along the thickness. The FGPM layer and two homogeneous piezoelectric layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. The Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on electric loading, gradient of the material properties, crack moving velocity, and thickness of the layers. The following are helpful to increase resistance to crack propagation in the FGPM interface layer: (a) certain direction and magnitude of the electric loading, (b) increasing the thickness of the FGPM interface layer, and (c) increasing the thickness of the homogeneous piezoelectric layer to have larger material properties than those of the crack plane in the FGPM interface layer. The DERR always increases with the increase of crack moving velocity and the gradient of the material properties.