  We describe a method to compute the Brauer-Manin obstruction for smooth cubic
surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is a 3-group. Our approach is
to associate a Brauer class with every ordered triplet of Galois invariant
pairs of Steiner trihedra. We show that all order three Brauer classes may be
obtained in this way. To show the effect of the obstruction, we give explicit
examples.
