In this work we use the Topological-Shape Sensitivity Method to obtain the topological derivative for three-dimensional linear elasticity problems, adopting the total potential energy as cost function and the equilibrium equation as constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to calculate the topological derivative. In particular, firstly we present the mechanical model, later we perform the shape derivative of the corresponding cost function and, finally, we calculate the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities. In order to point out the applicability of the topological derivative in the context of topology optimization problems, we use this information as a descent direction to solve a three-dimensional topology design problem. Furthermore, through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm.