I suppose I'm one of the irritating ones!! 
And me as well!
I do have to say that I've really enjoyed reading this thread and learning about everyone's perspectives. When I studied abroad in England, I really started to wonder what makes an American an American. How would I identify a fellow American as opposed to a British person, if we took away the accents? Is it shared cultural experiences and references?
It is the consequence of our country beginning its life as a smattering of European settlements and colonies a mere 300 years ago and hosting several waves of immigrants that part of being American, in my opinion, includes a collective acknowledgement of other nationalities and cultures in the makeup of this country. Individual Americans choose to acknowledge it in varying degrees, ranging from not at all to speaking the ancestral language and preserving ancestral traditions.
My pedigree is 100% English on my father's side -- they arrived in Massachusetts Bay Colony back in the 1630's. My mother's parents were more recent immigrants -- both from Ulster. It always amused me that my mother (a bit of a Hyacinth Bucket type) liked to think of herself as "Scottish" and maybe the family did come from Scotland generations ago. But my grandparents were Irish. They thought of themselves as Irish, their immigration papers gave "Ireland" as the place of birth, their parents were listed as "Irish".
And now I am officially Irish. Does that mean I can call myself "Irish-American" to designate the dual nationality? 
I understand this entirely, and it's something that I'm thinking about as I pursue Italian citizenship. My maternal ancestors were from Italy, although my family has been in the US for 100 years. So it's not like they got off the boat yesterday. But our heritage has always been significant to our family. However, I would not go to Europe and say that I was Italian. I would say that I was American with Italian ancestry. Yet, when I receive my Italian citizenship, I *really* will be Italian, even though I don't speak Italian and would feel like an American first if I was plopped in the middle of Italy.
It's worthy to note the Italian government's position through citizenship by descent. Unlike most countries that offer
jure sanguinis (rule of blood) citizenship, Italy does not limit the number of generations through which one can claim citizenship. Italy lost so many citizens to emigration and by establishing this policy, is in a sense trying to reclaim them. Furthermore, Italy holds the position that someone eligible for
jure sanguinis citizenship has really always been a citizen of Italy, and by going through the process is merely having that citizenship formally recognized.
However, the
jure sanguinis law was established way back in 1912, in the throes of mass migration. Now that several generations have emerged of Italians who don't speak Italian, who have never been to Italy, and have little understanding of "true" Italian ways, some Italians (as in people born and raised in Italy) want to have this law repealed.
