My ancestors all came from Eastern Europe from what is now Poland, Belarus/Lithuania, and western Russia. English was not the first language for three out of four of my grandparents.
By the time I came along, none of them had any foreign accent at all. It was very important to them to adopt the language and culture of their new home when they immigrated here and they worked very hard at speaking English with perfect fluency and no accent. They felt that opportunity, success and perhaps even survival were dependent on it.
I never once heard any of my grandparents speak anything but English (except for Hebrew during religious services). In fact, it never occurred to me that they even knew another language.
Apparently, that they might know other languages never occurred to my sister either. My sister is nine years younger than I am and when she was around four years old she was staying for a few days at our grandmother's house. My grandmother received a call (extremely rare) from family back in Lithuania, and since the other party didn't speak any English, she began speaking Lithuanian (her native tongue).
My sister completely freaked out. She had only ever heard grandma speak English and all of the sudden grandma was spewing forth what sounded like bizarre gibberish to my little sister. My sister got really scared that something was seriously wrong with grandma and started crying and it took a while to get her to calm down.
That shows the extent to which my grandparents completely rejected their heritage and completely adopted as many aspects as they could of the United States. You would never guess that they were immigrants.
The melting pot worked.
Now, the emphasis is on multiculturalism. Immigrants and children of immigrants are encouraged to keep their language and culture. They're not encouraged to assimilate. Member of their group, sorted by culture, first; citizen of the United States second (or possibly third behind religious affiliation).
We'll see where that takes us, but as we seem to become ever more polarized and fractured, I wonder if that's part of the reason.