Yeah you're right but taditions remain traditions. Italians have traditions that vary according to the place of origin and modern day italian-americans from America have close "ancestors" ( like grandparents or great-grandparents, since the emigration from Italy to America started in the 1920 ) who came from Italy and hand down their traditions.
You may say.. What kind of traditions? For example criminal organizations from southern Italy traditionally and apparently are very religious, they are people who are particularly attached to their parents and have particular uses which obviously, with time, mixed with American uses, but they still have relevance in my opinion for a correct interpretation.
This is not a "how to properly roleplay an Italian Mobster Guide" and I don't seem to have written it anywhere. This guide ( which as you can see from the title is still a work in progress ) is an incipit of how Italian organized crime evolved starting from the origin.