I have the non-accent accent Anglicized second-gen Canadian immigrants have, which basically could come from anywhere in Canada or much of the northern parts of the United States. I doubt it would have any special appeal, but I can enunciate most any phoneme or morpheme effectively and without affectation. It's not beautiful, but it's utilitarian.
I'm ashamed to admit this, but I lost 90% interest in a university course because the instructor had a thick Indian accent. I can't really complain about accented English, all things being as they are, but it still pisses me off when I need to understand someone and I just can't make it out what they're saying. A lot of recent immigrants get call centre jobs for instance, or work for help hot-lines and the like - asking multiple times for clarification is just painful.