In this paper a conceptually simple integral-equation formulation for homogeneous anisotropic linear elastostatics is presented. The basic idea of the approach proposed here is to rewrite the system of differential equations of the anisotropic problem to enable the use of the isotropic fundamental solution. This procedure leads to an extended form of Somigliana’s identity where a domain term occurs as a result of the anisotropy of the material. A supplementary integral equation is then established to cope with the resulting domain unknowns. Although the solution of these integral equations requires discretization of the contour of the structural component into boundary elements and its domain into internal cells, the numerical scheme presented here depends only on the boundary variables of the problem. Once the boundary solution is obtained it is possible to compute the unknowns within the domain, if required. The main objective of the present work is to develop an alternative integral-equation formulation that could be used to reduce the time needed to compute three-dimensional solutions for linear homogeneous anisotropic problems. Another possible advantage of the proposed formulation is its generality, which enables its direct extension to include dynamic and plastic effects in the analysis. Encouraging results are presented for four examples where structural elements are submitted to tension and shear effects.