The Teenage Mutant Ninja Turtles have been a pop-culture staple for over thirty years. The heroes in a half-shell have starred in all manner of media -- from video games to cartoons to live-action films. The 1990 movie adaptation was one the highest-grossing independent film of its time and also one of the final films that legendary puppeteer Jim Henson ever worked on. The film's numerous fans have speculated for many years about certain creative choices, such as why the Turtles have such distinctive accents, sounding nothing like their Master Splinter who raised them. One fan theory on Reddit provides a possible explanation.
In all three original film adaptations of the Teenage Mutant Ninja Turtles, Leonardo and Donatello have non-descript American accents, whereas Raphael speaks with a heavy New York accent and Michelangelo speaks with an equally heavy West Coast accent. This is strange, as it would stand to reason that the Turtles were taught to speak by their Master Splinter, who speaks with a Japanese accent. According to the fan theory, the difference in their accents is largely due to each of the Turtle's personalities.
While it may be true that Splinter teaches the Turtles the art of ninjutsu, most of the Turtles' verbal lexicon clearly comes from film and television. According to the theory, this had the biggest impact on Michelangelo, who picked up his Jeff Spicoli-esque accent and his penchant for creating memorable catchphrases from his love of the silver screen. Michelangelo also has a knack for impressions, showcasing his versions of Sylvester Stallone, James Cagney, Humphrey Bogart and others. The accent itself actually dates back to the earlier TMNT cartoon series which also highlighted Michelangelo's skateboarding ability, another tenet of West Coast culture at the time.
On the other end of the spectrum (and country) is Raphael's distinctive New York accent. The accent is explained in the theory by the fact that Raphael spends more time above the sewers by himself than the rest of the Turtles. Early in the film, Raphael leaves the sewers to go see a movie, something he apparently often does often to blow off steam. Evidently, the amount of time he spends immersed in the crowded New York City streets, and possibly his deep desire to be accepted by society, has manifested in Raphael emulating the vocal inflections of those who dwell in the city above him.
While Leonardo and Donatello both enjoy film, television, sports, music and other pop culture as much as their brothers, the remaining two Turtles' voices lack any distinct accent. The theory posits that Leonardo and Donatello make concerted efforts to sound neutral. For Donatello, this is due to his love of books (such as Tolstoy's War and Peace) and his need to be able to explain complex technical jargon to his brothers with as little confusion as possible. For Leonardo, his pride and role as de facto leader of the Teenage Mutant Ninja Turtles drive him to speak properly, even stopping himself from using slang when speaking with Master Splinter.
Though unconfirmed, this theory makes a relatively strong case for why Mikey and Raph have different accents from their brothers.