It still exists. Most people don't see any in themselves. Much of it is subtle (or stealth). Now, none of what follows does more than scratch the surface. Nor are there solutions contained here--though there are a couple of things that I think would help.
When I get on the 'L' in Chicago, I see all sorts of people. There is a peculiar phenomenon related to people seldom choosing to sit right next to someone they don't know. As the cars fill up, more of those seats are taken. People who are smartly dressed are among those sat next to first. Then those who look "clean," i.e. no stains, tears, groomed (shaved and brushed). Other "non-threatening" types fill up quickly too, this includes parents with children, elderly, many women, travelers. Some of the last seats to be occupied include those next to younger, street clothes dressed men, and the last of those black. But here's the question: is that really the way it is, or is that just the way I see it? If it wasn't for racial overtones, would I even begin to make mental notes about things like that? Does it affect the seat that I choose in a crowded car? If I deliberately choose one way, either way, is it good or bad?
I know, in fact, how I chose what seat to sit in when I must pair up: the seat I see first with the easiest access and most room that doesn't put someone else out. Of course, I'm not sure how I choose if two seats fit the bill. Even still, that doesn't answer the first two questions. Those are harder.
Here's another bone to chew on: affirmative action. I really don't have a problem with it. I'm not sure I would have a problem if it ended either. I generally see it as a good thing, but I know that there are people (white men) who see it, itself, as racist. This gets twisted into two distinct arguments, both of which I hate: 1. Affirmative action is anti-white; 2. Affirmative action is anti-black. The argument for the first is obvious, if wrong, the second is more seditious and nasty. The only real argument against affirmative action is that it is used as a talking point to stir up white anger (GOP voters, almost all) and it, therefore, is not really affirmative in practice, but that argument is weak as it is largely based on the prevalence of the other two.
Then there is the all-powerful: "I'm white, therefore my opinions on this are irrelevant." Which both assumes that all white people are racist and that fixing problems associated with racism is up to those who have felt the brunt of it, rather than those who, most often, have dealt it out. There is also a socialism/communism issue to be dealt with here most commonly reflected by the dishonest line: "Why should I have to be punished for the actions of others?"
Why it is that upper middle class white men so often feel persecuted is beyond me, but it has been a driving force for the Republican party for years, and it seems like it will continue in the near future (immigration, anyone?).
Racism is a very complex problem and it is one of many bad things that comes out of the self as victim perception that is popular in the US. That is to say that if a person feels they do not receive their due then they will want to find a person/group/thing to blame, and it is far easier to place blame on someone or some group that is obviously different, and race fits the bill. I generally believe that such feelings will be more prevalent in a society with more rigid class structure--like this country right now.
A few specific things would go a long way to fixing the problem. Improving education, including student aid, and providing first home purchase aid, and universal health care which really should be a given. I have plans; they don't belong here.