In my experience, most English teachers in the US spend almost no time teaching pronunciation. When they do, you can’t really talk much about methods, because they really don’t know what they’re doing. Phonetics and phonology are two subjects that most American students of education hate and try to avoid learning. (I know, because I used to have to teach those subjects to them.) Because most of them have learned very little about speech sounds, they have no principled way of helping ESL students learn them. There is the further problem that many American ESL teachers have been taken from other, nonlinguistic fields, and those have no linguistic training at all.
Frequently, because they don’t understand how pronunciation works, these teachers focus too much on individual speech sounds, and not on things like linking or prosody, which are very important in getting the sounds right.
Often, if an advanced student has very severe pronunciation problems, instruction will do him little or no good, because the root of the problem is his personality. It’s been mentioned in ESL literature (and I’ve seen it myself) that students with very severe pronunciation problems have very rigid personalities, are arrogant, resist change, and are excessively worried about their dignity. This makes them stay within their own phonological system even in other languages. The irony is that trying to maintain their dignity results in very undignified-sounding English. I should say that in my experience this does not apply to all students with severe pronunciation problems. Speakers of East Asian languages and Spanish seem to have more severe difficulty even if they’re humble and earnest.
American kids, by the way, often cultivate a bad accent on purpose when they’re learning a foreign language. They’re afraid that if they sound good, they’ll look like show-offs to their classmates or will appear to be sucking up to the teacher. In my own family I have a young relative who can pronounce Spanish with a perfect Latin American accent, if he is forced to, but instead has cultivated the habit of speaking it with such a terrible gringo accent that even I can’t understand him most of the time.