My school has a relatively small minority population, about 10% Asian, 10% Latino, 3% Black. Both NPC and IFC are still very diverse in their population, for those wanting the more traditional sorority or fraternity experience. And there's nothing wrong with joining one of those organizations. Two of my best friends and my boss are in NPC sororities. And I, along with some of my sisters did go through the NPC recruitment, but didn't find the "fit" we were looking for, so we went over to the cultural organizations. My campus has about 15 Latina/o, Asian, multicultural, gay, and religious GLOs on our campus, with a BGLO and lesbian sorority in the works, spread across the NPC, IFC and CGC. It's just a matter of finding where you fit. My sorority is a Latina Interest Sorority, but our newest class has members who are Vietnamese, Chinese, Scottish, and lesbian/bi, plus our other members who are Italian, Indian, and Fijian.
There's nothing wrong with joining an organization that isn't your ethnicity, and there's nothing wrong with joining an organization that doesn't specify a cultural focus. It really is about getting along with the people in the org, as well as getting the most out of your experience in that org.
Suggestion: If you want the Asian culture, along with the sorority, and didnt find your fit anywhere else on campus, you could always look into starting a chapter of an Asian sorority, allowing you to create the type of environment you want.
And if you still can't find what you're looking for, there's the professional GLOs.