This paper presents a direct boundary element approach for anisotropic static three-dimensional linear elastic problems. Formulation is based on the use of regularized boundary integral equation (BIE) for displacements. This BIE is weakly singular which is advantageous compared to the traditional strongly singular formulations. The displacement static fundamental solution is expressed in terms of an integral over a circumference with a unit radius. For the efficient numerical implementation of these fundamental solutions an interpolation scheme is used. For the spatial discretization a mixed approximation of geometry and boundary fields is employed. Numerical solutions of the problem of spherical cavity in an infinite elastic medium subjected to the uniform internal pressure are presented for the materials with different degrees of anisotropy.