My earlier post led to some very good observations. James asked for some examples about what I was talking about in saying that our American Catholicism is more Irish than we realize.
I don't have all my sources on hand, but just off the top of my head--
One really big way the Celts influenced Christianity (in the West, that is), is that their New Year's observances, with a look to the deceased and debts owed the dead, led the Church of Rome to move the "Feast of All Saints" from May to November 1, coinciding precisely with the Celtic New Year. Irish Christianity also impacted the whole Church in the practice of "confessions of devotion" for venial sin. This was part of the practice of spiritual direction in Irish monasteries, and when the missionary monks brought it to the continent, it was initially considered quite suspect! They even had little penance manuals indicating what sort of atonement matched which faults (and the penances were, by our standards, huge!). This eventually had a twofold impact. One, we know: the practice of private devotional confession was adopted and recommended by the universal Church as a valid use of the Sacrament of Penance. The other effect is less obvious, but the practice led to a greater "interiorization" of the spiritual life. The focus on virtues and vices and sins moved to the interior realm of thoughts and impulses. Prior to the Irish innovation, the "canonical penance" of the Church Fathers was generally used for sins that were of a public nature and haad a demonstrable effect on the community. This "new" practice had the unintended side-effect of spawning scrupulosity as a spiritual disease. Thus in a general way for the universal Church.
One surprising source of examples of Irish influences in American Catholicism would be the book "Why Catholics Can't Sing" by Thomas Day. Besides music, he covers some of the history of American Catholicism (since for 200 years, the majority of our bishops were, in fact, Irish). But the musical part is impressive!
Some general characteristics of Irish-American Catholicism include the "privacy" of our piety (a kind of spiritual reticence, perhaps a relic of centuries of persecution); a leaning towards Jansenistic tendencies (I can't remember if it was the Irish priests getting their training in Jansenist-era France, or the persecuted French priests escaping the Revolution and bringing Jansenism to Ireland)--at any rate, the Irish picked up on the rather gloomy Jansenist spirituality much more heartily than the French seemed to have; and, according to Fr. Benedict Groeschel, a repression with regard to sexuality that is entirely foreign to the Catholicism of southern Europe.
Now that the US Church is 30% Hispanic, we have to expect some cultural shifts--it would be quite misguided for us to force Latino Catholics to become Irish, thinking that this is "real" Catholicism. It's a challenge for all of us!