Actually Sulu was meant to be of nonspecific, pan-Asian heritage. His "official biography" in The Making of Star Trek
characterized him as "of mixed Oriental and Filipino descent." The name "Sulu" doesn't exist in Japanese; it's the name of a sea in the Philippines. The only reason Sulu's come to be considered a Japanese character in retrospect (and given the first name Hikaru by novelist Vonda McIntyre in 1981) is because he was played by a Japanese actor. (And maybe because he imagined a samurai in "Shore Leave" -- but then, he also fantasized about being D'Artagnan in "The Naked Time.") But of course, the example overlooked in the first post is Korean actor John Cho taking over the role of Sulu in the current film series.
And let's not forget Chinese-American Rosalind Chao playing Japanese Keiko Ishikawa. Or, conversely, Japanese-American Brian Tochi playing two different
Chinese-surnamed characters: Ray Tsing Tao in "And the Children Shall Lead" and Kenny Lin in "Night Terrors."
I'd say the most prominent Asian performer in Trek who played her own ethnicity was Patti Yasutake as Nurse Ogawa. Other examples of Japanese actors playing Japanese characters include Miko Mayama as Yeoman Tamura in "A Taste of Armageddon" and Clyde Kusatsu as Admiral Nakamura in TNG.
There is the ambiguous case of Korean-American actor Daniel Dae Kim playing MACO Corporal Chang in Enterprise
; Chang can be either a Chinese or Korean surname. (Although the Star Trek Customizable Card Game
gave Chang a Japanese first name, Hideaki.)