I would argue there's some work that needs to be done on the other side of the coin, too. There are too many people making money off feeling oppressed. I personally don't feel the answer is making honest people go out of their way to carefully word their sentences to avoid mentioning a characteristic of a person, I feel the answer is understanding that saying "that is a person" is nowhere near as descriptive as "a black man from Chicago".
And herein lies the difficulty with reconciling language (which is a way to communicate descriptive experiences) with political correctness. Is his race relevant? Is his gender relevant? Is his city relevant? Maybe, maybe not. But the point is to describe the scene as it exists. The mere mention of descriptive words are often taken as prejudice when that's not the intention. Let's flip that back around. When I say "that person from Chicago", who do you picture in your mind? If you picture a man/woman, black/white/latino/asian, tall/short/fat/etc, you're being "prejudiced". Because I never mentioned any description of the person. But without that information, your mental picture of the scene is incomplete and your brain is going to fill in those blanks. Language has words you can add or omit to imply discrimination. "That black man from Chicago." "Another Asian from Chicago". "Of course, he's white." In absence of these deliberate hints, it's asinine to assume anyone is taking any sort of position.
When I go camping I don't say "I stayed somewhere with plants", I say "I slept in my tent on the ground in the Yosemite forest." That's not prejudiced against trees just because I called it a forest, nor against the Earth because I mentioned the ground. The more prevalent problem is not people being prejudiced, it's people feeling prejudged just because they expect to be prejudged. Why should someone go out of their way to reword and complicate their story when you're (hypothetical "you") are going out of your way to unword and read more into their story than they actually said?
(This isn't in response to the linked story, merely to mistercow's point)