I think that the thing we "can't get our heads around" is the fact that you have a nationality (American), your parents have the same nationality, your grand parents also. Whereby comes the need to look for something else?
The answer to this question is that nationality does not equal culture. Nationality is in many ways irrelevant in that context. After all, I share the same nationality with people who look completely different from me, speak a different language, practise a different religion, and have completely different values and ways of thinking.

Surely, living in France, you see similar things there. If you have children born in France, would you expect them to be "French" in the same manner as the large contingent of those who came there from the Middle East? Again, I'm talking about culture, values, manner of dress, religion, etc. While your French children may share a common nationality with others, that doesn't mean that they are expected to be automatons. Their culture and values come from their family, and all the ethnic history of that family. It would be a sad day indeed if your children, as well as the children of those immigrants from the Middle East, intentionally forgot where they came from and who they were!

I'm slightly disturbed by the modern trend of associating culture with nationality. Nationalism really has no place in it. This is 2012, for crying out loud. It's the age of globalism, and to expect that everyone who lives within the imaginary lines drawn on paper by politicians should rigidly adhere to a similarly imaginary culture that defines that nation is a trifle absurd, don't you think?

It is, in fact, the very phenomenon of globalism (i.e. the spreading of peoples from their ancestral homelands to points all over the globe) which makes it that much more important to attempt to retain some semblance of cultural identity. By preserving the culture of one's forefathers, an individual is resisting the push towards the horror of assimilation and homogeneity. By preserving culture, we implicitly state that the culture has value and should not be forgotten. What a cheap world it would be if we all abandoned our family history and traditions in favor of shallow modern culture!