In the present paper, a new version of the elastodynamic finite integration technique for axisymmetric wave propagation in homogeneous and heterogeneous linear elastic media (CEFIT) is presented. This special variant of a finite difference time domain scheme offers a suitable method to calculate real three-dimensional problems in a two-dimensional staggered grid. The implementation of boundary conditions at inner and outer boundaries of the model is much easier than in standard FD schemes using non-staggered grids. In order to test the accuracy of the numerical CEFIT code, problems for which analytical solutions are available are presented. These solutions involve wave propagation in an elastic plate and plane wave scattering by a spherical obstacle. Other applications of more practical interest are modeling of compressional US transducers and ultrasound generation caused by a thermoelastic laser source.