Transient dynamic behavior of a three-dimensional (3D) homogeneous cantilever beam under sinusoidal loading at the free end is verified using the finite element method (FEM). Explicit central difference technique is used for the time integration of finite elements. The tip displacement and maximum stress at the fixed end obtained using the FEM agree well with exact solutions. Modal analysis of a functionally graded (FG) 3D cantilever beam is investigated using Rayleigh-Ritz (RR) method and the FEM. The natural frequencies obtained using the RR method converges as the number of terms in the assumed base function increases. The natural frequencies vary considerably with the gradation of the beam, more for lower modes than for higher modes. Wave propagation in a fixed-free 3D bar is studied using the FEM. Axial stress results for the homogeneous bar with zero Poisson’s ratio agree closely with 1D exact solution. For the FG bar, we see that gradation affects stresses considerably more so at the fixed end than at other locations.