I've used them and my clients and the customers of those clients were happy with them. I did not through paragraphs of text on them, just pretty images and headlines (featured content).
Even if you did this, your mobile users will still be mad, because they cannot hover.
The question isn't if you or your clients are happy with them. The question is if your users are happy with them, and if they generate better business results (conversion, clickthrough, whatever is relevant to you) than the simpler, less Javascripty, less timing-based, less gotcha-with-the-mouse-hover implementation.
I sincerely hope you A/B tested this.
Though some have a habit of stubbornly ignoring any professional advice against a feature they have an emotional attachment to, raw data can sometimes snap them out of it.
The second terrible mistake is using page indicator dots (which do a good job of indicating the page) for navigation, let alone using them as the only means of navigation!
When you take into account these two things, a carousel can actually become comfortable to use and may no longer enrage its users. But that's still a far cry from being better than showing the content in a regular list:
> I've used them and my clients and the customers of those clients were happy with them. What do you mean with "happy"? Did the customers actually understand how to use the carousel, did they discover its content, did they click on it? Not being annoying does not suffice to qualify as useful.
Happens most often on newspaper and magazine websites. Some sites provide a pause button to kill the distraction, but many don't.