The propagation of axisymmetric electroelastic waves in hollow cylinders made of a radially polarized functionally gradient piezoceramic material is studied. The material properties vary across the thickness with a power law. The lateral surfaces of the cylinder are free of loads and covered by thin electrodes to which alternating voltage ±V0 exp[i (kz−ωt)] is applied. To solve the problem, the efficient numerical-analytical method is proposed. The original partial-variable three-dimensional electroelastic problem is reduced, representing components of an elasticity tensor, components of displacement vector, electric-flux density, and electrostatic potential by running waves in an axial direction, to a boundary-value eigen-value problem for ordinary differential equations. This problem is solved with the stable discrete-orthogonolization method. The results of the numerical analysis for the cylinder made of a functionally gradient material (metal and PZT 4 piezoceramics) are presented.