Please note special day and time.  We discuss the universal triviality of the CH0-group of cubic hypersurfaces, or equivalently the existence of a Chow-theoretic decomposition of their diagonal. The motivation is the study of stable irrationality for these varieties. Our main result is that this decomposition exists if and only if it exists on the cohomological level. As an application, we find that a cubic threefold has universally trivial CH0 group if and only if the minimal class θ4/4! of its intermediate jacobian is the class of a 1-cycle (only twice this class is known to be algebraic).