Nowadays, sure, but keep in mind that the "US" didn't extend beyond the east coast when the Puritans first settled here. You might be able to make an argument that there's no cultural influence from the colonial days that lasted until today (although I'd disagree with that sentiment), but otherwise, where would you expect any cultural influence in the rest of the US to have come from?

(To be clear, I'm not saying that there weren't existing cultures there before the US expanded out further west, but I imagine most people would agree that the US today isn't culturally as influenced by them as much as from the the colonies and pre-expansion US.)