This is not quite accurate. In fact, SCOTUS has walked a fine line on this question, never definitively settling the question of whether "the people" refers to the citizenry of the US or people in general. For example, non-citizens were found to be partially protected by 4th ammendment prohibitions regarding search and seizure in the late 60's. The court's reasoning for this, and the limits of partial protection were extremely murky. It is my belief that if the current, more strictly constructionist, SCOTUS were forced to decide the issue you would have a good chance of establishing that constitutional rights apply exclusively to citizens of the US.
Well I will certainly say that citizens have certain "rights" that are denied non-citizens. Running for and holding public office comes to mind. As does the purchasing of (some) firearms though those may still be possessed (In fact some states will allow foreigners to purchase rifles and shotguns). Foreigners commonly hunt, for instance.
However non-citizens may pratice freedom of speech, association, religion and the press. They do not have to house soldiers. They are secure in their papers. Miranda applies. As does "speedy trial", impartial jury, Grand Jury, "double jeopardy", "self-incrimination", freedom to travel, "right to an attorney", and "search and seizure", excessive bail, excessive fines and cruel and unusual punishment.
While there are probably some attorneys out there "picking nits", in general, non-citizens have the same protections as citizens.