S.H.
I can answer this from two perspectives: I am the child of immigrants and I have raised a child in yet another country/ culture/ language.
As a mother I found that it was fundamentally important that my child be fluent in two languages. We spoke English in the family but the native language around everyone else. She's still fluent in both.
I purposely had my daughter overseas so that she would be a dual citizen and if war is to ever affect her in her life, she could flee to a safe place. We were living in the country where she was born anyway, but the option to birth here was also available. We've seen too much of the world to think that war is far from even the US, so we wanted her to have options.
Anway, there were great things about that culture, from our perspective, and things that we strongly disagreed with. We disagreed with the fact that we did not know a single couple who was faithful to each other. We were involved in our community, we have good friends of the other culture, we're not being judgemental. They'd tell us, too, that it wasn't a big part of their culture. Consequently, HIV numbers are staggering in that area. So we were pretty clear that by the time our daughter was of dating age, we'd be gone from there. As the child of an immigrant, I know that you can't take your values from another country and expect them to be followed in your kid's home country. We weren't going to fight that battle, so we left. We were lucky to have that option.
We moved back to the US, our birth country, and my dd is a US citizen by birth, though she was born on foreign soil (and actually, yes, she can become president one day, but her kids must be born on US soil in order to retain citizenship. If, however, she adopts, those kids will be US citizens when they touch the ground here. That might be TMI.)
Her personal identity is that of her birth culture, which is interesting because she has not a drop of blood from that area. She sees herself as much that as what she has inherited from my husband and I. I feel like our job is to nurture her inner compass, so we tell her our perspective on where she comes from, but encourage her to find her own identity. I have always hated hated hated when people said that I should be more _______ because that is where my parents come from. I also resent that I should be more like the folks who've been here for many generations. That isn't me. I"m a medly of all these experiences and no one has the right to define me for me. I'm a work in progress, for sure, but I'm MY work.
My parents had a really hard time with a lot of the cultural difference. They had an arranged marriage, and though they would have liked that for me, they didn't even attempt to force it. I would have left and they never would have seen me again. That's at the heart of the issue for me, you know? You have to be careful in instilling identity without alienating your child.
I ended up being a lot more American than the kids of my parent's friends, but they are perfectly who they are.
I think that teaching culture, tradition, language, food, values, pride is all very important, but it is also very important to recognize that if you are raising kids in America, then they are and will be American. If you insist on raising, say, a German and not an American, then it is a good idea to go back to Germany. My parents regretted, for a time, not taking us back to the motherland to be raised since they disagreed with how we acted as teens. Things are much better now. They see that we are happy, but I think my M. especially hurts that I left their faith and I don't have a connection to their community. My brother is more traditional than me.
I guess you have to decide what is really important and what you're willing to compromise on and then decide where/ how to raise your kids.