Free and forced vibration analyses of functionally graded hollow cylinders and spheres are performed and analytical benchmark solutions are presented. The material is assumed to be graded in the radial direction according to a simple power law. The Laplace transform method is used, and the inversion into the time domain is performed exactly using calculus of residues. The Complex Laplace parameter in the free vibration equation has directly given natural frequencies, and the results are given in tabular form. On the inner surface, various axisymmetric dynamic pressures are applied, and radial displacement and hoop stress are presented in the form of graphs. The exponent in the power law, called the inhomogeneity parameter, essentially refers to the degree of inhomogeneity. Increasing the inhomogeneity parameter provides a stress-shielding effect. Closed-form solutions obtained in the present paper are tractable, and they allow for further parametric studies. The inhomogeneity constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution.