From "The Local-Italy" https://www.thelocal.it/20210723/can-tourists-use-italys-covid-health-pass-to-access-museums-concerts-and-indoor-dining/?order=4 .
It might be paywall protected so here are a couple of highlights pertinent to Americans.
At the border, Italy accepts vaccination certificates, tests results and medical certificates of recovery from the United States, Canada or Japan. That includes paper vaccination cards from the US bearing the CDC logo.
However, the government has not yet confirmed whether these documents will be accepted at venues within Italy once the green pass is more widely required.
My guess is they will figure it out as France has done, that the CDC card will be accepted in restaurants. If not, I guess well stay in Switzerland where we are starting our trip.