"Is ethnicity still important to white ethnic racial groups in the 21st century?"

I think that to a degree it is still important, but on a more personal level with individuals. Since people who are cast as "white" or  Caucasian, an individuals background or ethnicity ins't as upfront as it was 100 years ago. In the past, someone who was white or Caucasian may have been labeled more with their ethnic background. Instead of "white," you may have been Irish, German, or Polish. In the world today, if you were to ask the same question, most would probably say white or Caucasian. 

When I say it is more personal in todays world I am relating it to family history. This is even important to me because when I've been told about my family history, I am always told of our strong German background and that relatives are still continuing the family tradition back over in Europe. I think this same idea applies to others whether they may have great grandparents who immigrated from Poland, Germany, Ireland, or even Russia. So is it still important? In the regard of family history and background, I say yes.