We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for E/F a quadratic extension of p-adic fields the associated unitary group G=U(n,n) contains a parabolic subgroup P with Levi component L isomorphic to GLn(E). Let π be an irreducible supercuspidal representation of L of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation ιPGπ is reducible.