Her grandparents have told her that they don't want anything to do with her if she gets involved with one of them, and that she'll be disowned by her family if this behavior continues. She doesn't seem to care, even though her heritage is important to her, too. How do we get her to understand that we want to preserve our culture?
What she doesn't understand is your family's lack of tolerance for her liking people other than those in your ethnic group.
If your family continues to try to shame and threaten her for associating with other ethnic groups, then you will create more and more emotional distance between your daughter and yourselves.
It's up to you to make your choices on what you value the most. I hope that your love for your daughter will help you open your hearts and minds to the fact that she is not dishonoring anyone by choosing to be friends with or date others outside her ethnic group.
If you want some help with this problem, I'd suggest that you consult a family therapist. Everyone will get a chance to voice their opinions with a therapist in a respectful environment.