The equations governing three-dimensional elastodynamic scattering from planar cracks are formulated and solved in the frequency domain by Boundary Integral Equation (BIE) methods. The formulation requires a regularization of all nonintegrable kernels in the representation integral for the scattered stress field. The regularization procedure is novel in that it requires an initial discretization of the crack. The resulting discretized system of integral equations can be solved explicitly for the unknown crack-opening displacements. The crack-opening displacements, in conjunction with the appropriate representation integral, have been used to calculate far-field quantities of physical interest. Numerical results are compared with those from earlier papers dealing with a penny-shaped crack under normal incidence of a longitudinal wave field. The code has also been applied to elliptic crack geometries of various aspect ratios under normal longitudinal wave incidence. Numerical results are given for crack-opening displacements, scattered far fields, and scattering cross sections. The numerical results indicate that at sufficiently high frequencies, the scattering process is significantly affected by the extra length parameter of the elliptic crack.