Given that the movie takes place during the reign of Richard I, all the characters should be speaking Early Middle English, something of a transitional period between Beowulf and Chaucer. With this in mind, the other actors' modern English accents aren't really any more "correct" than Costner and Slater's American ones. (And it'd be nearly impossible to duplicate an Anglo-Norman accent anyway, since the language probably didn't sound much like modern French at all. Why try?)
Many linguists theorize that the American accent actually is the original British accent.
Furthermore, as part of the Anglo-Norman aristocracy, Robin may not have spoken English: Norman French was the language of the English nobility of the time, and in fact Richard himself didn't speak English at all.
The Sheriff and his cousin, Guy of Gisbourne, both eye-rolling when the Bishop asks for God's blessings on the Sheriff? The Sheriff is a devil worshiper (and so is his cousin, possibly).
In the extended version of the movie, we learn a little about the Sheriff of Nottingham's parentage. The witch Mortianna herself was his mother, and she killed a baby in the palace so that she could substitute her baby in his place. This brings up a number of horrific implications.
Probably one of the most subtle is shown when Mortianna talks about how she has been seen as a freak all of her life. Given that, how did she become pregnant? Considering her lack of sympathy for Marian's plight, it is likely that the Sheriff is the product of rape, perhaps even rape by a nobleman... in which case, having her son produce the heir to the throne by raping a nobleman's daughter is the ultimate act of revenge.
This also explains why the Sheriff, upon demanding a "proper" wedding to Maid Marian, snarls that "I will not take her until we are properly wed! For once in my life, I will have something pure!"
This is also Fridge Brilliance as well. Had the Sheriff simply raped Marian without being married in a Christian ceremony first, the resulting child would simply be a bastard, and therefore would not be eligible to ascend to the throne. The pair's plan all along was for the Sheriff to marry Marian, father a son, kill her off, and rule as regent in his son's name.