I am just posting my comment as another, rather contradictory answer.  It <I>might</I> indeed be possible both to parameterize rational points and even prove weak approximation in an easy, geometric way, in spite of the theorem of Clemens-Griffiths that all smooth cubic threefolds are irrational.  This is because the more relevant geometric property is "stable rationality" (or perhaps something even weaker).  Weak approximation holds for all stably rational varieties.  So far nobody has been able to either prove or disprove stable rationality of cubic threefolds (the technique of Voisin using "decomposition of the diagonal" has so far been unable to resolve this problem).