There is, apparently, some justification for Kevin Costner's lack of accent in Prince of Thieves.
Nowadays, not a lot of dialogue is shown on film just as the actors speak it during filming. Much of it is overdubbed later, so that the voices are clearer. Costner was, apparently, told that he could read his lines without an accent during filming, because he would be able to redo them all in the sound studio.
Then, in an effort to get the film out (there was a smaller production of a Robin Hood film being made at the same time, and PoT wanted to be out first), the studio skipped the post-filming sound work.
So, the lack of attention to accents is, at least in the main, attributable to the studio rather than Costner. Of course, in better productions, such as The Lord of the Rings, studios bring in speech coaches to ensure the actors have the right accent. They don't just go, "Sound English, OK?" when there are gazillions of very specific types of English accents to choose from. If Costner was given no direction at all, I can't quite blame him for the whole mess.