As I said in an earlier comment, Tolkien was a linguist and as such was extraordinarily sensitive to linguistic nuances like accent and the effect of social class on speech. If you render the common tongue as English, and keep in mind the history and social status of the various characters, choosing an accent becomes pretty obvious.
Actually if you wanted to Americanize LOTR, the Hobbits would have Southern accents (country bumpkins), the Rohirrim Texan accents (close to the Hobbits, still country but a little more refined), and the Gondorians a neutral General American/ Received Pronunciation accent (educated middle/upper-class).
I'd give the Elves a French accent (refined and a little snooty) when speaking the Common Tongue. Quenya played the role of Latin in Middle Earth (dead language used for formal purposes), and Sindarin was an everyday language evolved from it, so a Romance language would be the closest social analog to it. To an American listener a French accent would best convey the extreme refinement and cultured history (not to mention snobbishness) of the Elves. If you wanted to get even more specific I'd give Elrond and the Rivendell elves a French Canadian accent and the Galadhrim a Parisian accent. Linguistically, a Welsh accent would be most appropriate, as Sindarin was patterned after Welsh, but it just doesn't have the same social/class implications that French does.
If anyone had a Brooklyn accent, it would be a Dwarf. Tolkien explicitly equated the Dwarves with the Jews, and based Khuzdul on Hebrew... so a Brooklyn accent would be extremely appropriate for working-class Dwarves like Bifur, Bofur, and Bombur. Dwarvish nobility like Gimli and Thorin would have a milder, upper-class Jewish accent.