The classic version is "no circulars" in the UK, but it's totally ineffective, as there's no legal backing for it.
Advertising flyers - frequently glossed as "pizza leaflets" from the endless supply that pizza firms stuff through every letterbox in their delivery area, seemingly daily - are a common bane of life in the UK, and there's no right to stop them being delivered.
In the US, the issue doesn't arise in the first place, because they don't have letterboxes in the way Europe does. European houses have a "letterbox" (though there is often no actual box) on the front door, and anyone can put something into it. Leaflets from politicians and from local advertisers are common. Newspapers are also usually delivered this way to subscribers. All sorts of delivery services, including the mail, deliver letters and small parcels (e.g. CDs, DVDs, paperback books) this way.
In US, though, you have a "mailbox", which is at the roadside, not on the doorfront. These are the classic ones on a post with a little red flag to indicate that there is new mail. Only the US Mail and the resident can put things in - and the resident can put letters in for the US Mail to collect, unlike the European system of only being able to put outgoing post into a pillarbox. It's a criminal offence to interfere with anyone else's mailbox, for the same reasons that it's a criminal offence to intercept or interfere with the mail in most countries.