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Suppose I'm talking to a person who is an American citizen but obviously has some "Asian roots" (either they immigrated to the US at early age or their parents were immigrants). Is there a polite way to ask about their ethnicity (if that's the right word)? If I just ask "Where are you from?", they will definitely say they are from the States since they grew up here.

First think: "Why do I need to know?" Just being curious is not a need to know. It is impolite to ask for personal information just to satisfy your curiosity.

If you don't need to know, then don't ask.

If you decide that for some reason you do actually need this information, then explain your reason and ask directly.

Don't ask. If you know someone well enough to do so, you will already know the answer. Considering there have been people of Asian/Chinese origin in the USA for more than 200 years, the person's roots in the US could be older than many Euro-ethnic Americans. There is no "polite" way to ask this question. You will always be saying "You look like a [slur]. What kind of [slur], exactly, are you?"

To the very good answers you've already received, I'll add that you're right to say that trying to elicit information about an Asian American's background by asking "where are you from?" isn't good idea. However, the problem isn't just that the person is likely to respond by (correctly) telling you that s/he is from the States (or a particular US state), as you've noted; it's also that many Asian Americans have been asked "where are you from?" over and over and over throughout their lives by people who really mean "where were your ancestors from?" No matter how benignly it is intended, to the person being asked, the question often feels both prying and duplicitous – and that's because it is. In fact, if you actually want to know where in the United States an Asian American is from, it's probably best to say something like "what part of the country did you grow up in?", so that you won't be misunderstood.

Finally, a question you might want to think about: When you meet Americans of European ancestry, how often do you find it necessary to ask them what country their ancestors came from?