A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.