A characterization of linearly reductive groups by their invariants

The theorem of Hochster and Roberts says that, for every module V of a linearly reductive group G over a field K, the invariant ring K[V]^G is Cohen-Macaulay. We prove the following converse: if G is a reductive group and K[V]^G is Cohen-Macaulay for every module V, then G is linearly reductive.