A stable node-based smoothed finite element method (SNS-FEM) is introduced to study the propagation of elastic waves in two- and three-dimensional periodic systems. In the SNS-FEM, three-node triangular and four-node tetrahedral elements that can be generated automatically are first employed to discretize the problem domain in 2D and 3D spaces, respectively. The system stiffness matrix is then computed using the smoothed strains over the smoothing domain associated with the nodes of elements. To cure the numerical instability and improve the accuracy, a stabilization item by considering the strain gradient is also taken into account when computing the band structures of the elastic waves in 2D and 3D phononic crystals. The band structures of different material combinations, different filling fractions, various lattice forms and scatterer shapes are computed. The results are compared with those calculated with the well-known finite element method (FEM) and the plane-wave expansion (PWE) method, where the advantages of the SNS-FEM are pointed out and analyzed. These advantages show that the present methodology is an ideal alternative for band structure computations in both two- and three-dimensional phononic crystals.