For the record, I am an Asian American woman, married to a white (immigrant). Growing up, I was all about the Asian guy--personally, I find them to be the most attractive men out there. Still do. There are unattractive Asian men, of course, just like there are unattractive men that are white, black, etc.. I definitely see attractive men of all colors, but I like Asian men best. Unfortunately, all the Asian guys I liked didn't like me back for whatever reason.
I grew up in a diverse area of the US, and I only knew a few Asian girls that explicitly said they preferred white/non-Asians to Asians. I can tell you my anecdotal evidence: almost all the Asian guys I grew up with are married and they're married to Asian women. I'd say about half the Asian girls I grew up with are married--some to Asians, some to non Asians (two Hispanic guys, but mostly some shade of white). The two girls that explicitly stated they prefer non Asians are still single (for whatever reasons). We are in our mid 30's. The single women are very high up the corporate ladder with very nice careers and are highly educated (lawyers at big firms, VP's etc..) so that may play a part in relationship status. Most of the married women are stay at home moms/low level jobs, though there are a few doctors (they're married to Asian doctors).
I want to add: most of the married guys and girls I knew were immigrants while the single women were born in the US. I think that matters--if you're more connected to your culture, you're more likely to find someone in your culture to marry. I was born here and was told I was "too American" by one of my Asian guy friends. That may have been why my Asian crushes never worked out for me, but who knows. My born in the US family/cousins also all married other Asians (most of the women were immigrants, but came over at young ages).
I can tell you in my very limited experience that the three Asians that did court me were very soft about it. They'd befriend me and then casually try to ask me to hang out or they'd write me a note (yes a note! Although, this was highschool), but would drop me as a friend once I said no. One of them stalked me and sent me an email out of the blue (when I only shared my email with a few people. This was in the 90's). The two non Asians I met were more aggressive but nonchalant, if that makes sense. They'd either chat me up and ask me out, but when I'd decline, they'd still offer to hang out with mutual friends, etc.. It seemed more casual and it felt much easier to say yes/no. I felt like there wasn't any bitterness after I said no. Maybe they were better at hiding it, I don't know. Sidenote: I also had the "creepy white guy approach" of stepping into my personal space and ask me if I was Korean/Japanese/Chinese with intense eyes, which I ran far, far away from.
The lack of bitterness/intensity of the approach could also be chalked up to maturity as the Asian guys that asked me out was when I was in highschool while the non Asians was after college. But if I were to give advice, I'd say a person has to be confident in himself/herself and mature about it. I have asked a few people out in college and been denied. It hurts! Especially as a girl who has been taught that "all guys would say yes to a girl," but the more "no's" you get, the more you realize it's not the end of the world. My world felt shattered for a few days after I was rejected, but then I realized the world didn't disappear. I moved on.
We each have worthy qualities and if someone you're interested in can't see them, well it's time to look elsewhere.