The US Supreme Court has said that you must invoke your right to remain silent. It doesn't seem constitutional, but it is currently the law. (Salinas v. Texas and People v. Tom)
That's not entirely true. They ruled that silence in a non-custodial interview could be admissable at trial. Silence in custodial interviews is still not admissable and they didn't actually decide the question of whether or not silence can be taken as substantive evidence of guilt.