(By “stereotypical,” I mean the one at about 5:32 in this video: The English Language In 24 Accents - YouTube).
Now, I haven’t met or spoken to a lot of people of Italian origin in my life, but of the ones I have spoken to, they have not had that stereotypical accent.
I would guess that the young man in the video is trying to emulate a Southern Italian or Sicilian version of an accent. He at times sounds like some of the characters in the Godfather films, which featured many real Sicilians. For a more typical Standard Italian-English accent, try Roberto Benigni, here: http://www.youtube.com/watch?v=ewrZlfCY8hU
Although many, if not most well educated Italians now speak Standard Italian, in many areas of the country the regional dialects can still be heard in wide use, sometimes by preference, sometimes by tradition and sometimes due to simple pride in the local identity and culture. Someone who spoke primarily in one of those dialects of Italian would probably speak English with a somewhat different accent than a Standard Italian speaker.