In the Stack Exchange voting system, we have two classes of votes: standard up / down votes that anyone can cast, and special accept votes that only the asker can.
Although accepting an answer does not always pin it at the top of the answer list, that green check mark does send a strong signal about the correctness of the answer. This gives a single person much greater weight than the rest of the community. That green check mark can seem confusing to visitors, who see potentially wrong or much lower quality answers marked as being correct. I've heard this from several friends recently who were surprised when they came across factually wrong answers that had been downvoted, yet still had that check mark next to them.
Moderators regularly field flags from even more experienced users who demand that the accepted answer be changed from one that is incorrect or even harmful to another. We decline these, because we're not supposed to be making technical judgments like that and because we have no way of accepting answers on behalf of the asker. However, these flags do show that some people have problems with the presentation of accepted answers vs. the community voting on all the others.
There's also all the fun that came with arguments over accept rate, and people badgering askers to accept their answers.
Questions on the Stack Exchange network are intended to be broadly useful, to help out many others in the future beyond the person asking the question. If that's our goal, why should the person asking the question have any more say than the rest of the world? If it's a broadly applicable question, why should one person be able to highlight something as correct simply because they asked it first?
I guess what I'm asking is: if we did away with accepted answers, converting them to upvotes where appropriate, and only let community voting sort out the best answers, would we lose something important? I thought I'd ask after reading Rosinante's answer to this recent question, because I wasn't sure how I felt about accepted answers anymore.