I guess they thought pausing four minutes between each word was Indian enough. Now that I mention it, I don't think he was Apache or a chief. His name doesn't make any damn sense. That's like putting a white guy on a team of Native Americans and naming him "Minnesota President."
--Seanbaby, on Apache Chief.
The concern with voice appropriation, if taken as an absolute rule, limits creative efforts to autobiography. Taken as a general caution against writing about matters of which one has no experience-- that is another matter. Talented male writers from Ernest Hemingway to Robert Heinlein have tripped over their own limitations when depicting the female psyche, while an entire cable network could be filled with Caucasian-generated misrepresentations of non-Caucasian people.
For further proof, consider DC Comics' well-intentioned past attempts to ethnically diversify the DC Universe.
A few one-shot ethnic heroes appeared in the late 1960s and early 1970s, but even these were mostly non-American white guys. DC's first real attempt to demonstrate that not all superheroes have fair skin begins with Saturday morning's Super Friends, which featured the adventures of DC Comics regulars Superman, Wonder Woman, Aquaman, Batman, and Robin. Other DC superheroes made appearances, along with some made-for-tv youthful sidekicks.
At some point, someone decided to introduce ethnic diversity to the Friends.
On the surface, this seems like a good idea. Sure, Superman is an extra-terrestrial, Wonder Woman hails from Themyscira (aka Paradise Island), and Aquaman grew up in Atlantis. Nevertheless, their all-American white bread nature seems clear enough. Why not have some other people demonstrate their world-saving abilities?
Enter Apache Chief, Samurai, El Dorado, and Black Vulcan.
Each has a name which relates to his respective ethnicity or race, rather than his super-power. Apache Chief can change sizes. As online wise-guy Seanbaby notes above, calling him "Apache Chief" makes as much sense as calling the token white guy "Minnesota President." Samurai dressed like a Samurai, spouted martial arts nonsense, and had the ability to turn himself into wind and fire. El Dorado, named ludicrously for a legendary lost city, had vague powers and a Mexican accent. Black Vulcan had lightning powers, like the very similar mainstream DC Comics character, Black Lightning. (Well into the 1980s, Afro-American superheroes generally had names which referred to their race).
Still, DC recognized the value of multi-ethnic heroes, and so the Super Friends comic tried to make amends, by introducing a whole new group of multi-ethnic metahumans, the Global Guardians, beginning in issue #7. A mystical Dr. Mist founded the international peacekeeping force, whose members are associated with specific countries. Japan had Rising Sun, who possessed solar powers-- just like DC's other Japanese heroes, Sunburst and Dr. Light. Apparently, a tie-in with the "Land of the Rising Sun" cannot be resisted. Other heroes' powers connected with the local weather; Icemaiden hailed from Norway while Fire blazed out of Brazil; she later went by the names Green Fury and Green Flame. Ireland was given Jack O'Lantern, who possessed a magic lantern given to him by-- of course-- a Celtic faerie queen. England had Godiva, a swipe of Marvel Comics' Medusa, with living tresses. Taiwan's Thunderlord voiced a seismic-shock-creating shout (another rip-off, this time of Marvel's Black Bolt). Australia's hero naturally called himself the Tasmanian Devil, since "Kangaroo" powers would have looked silly.
The Greek Olympian wore the legendary golden fleece, which gave him the combined powers of Jason's Argonauts. Tuatara from New Zealand, like El Dorado, boasts an indigenous name unconnected with his power, in this case the ability to see into time*.
The U.S. had a Native American representative, Owl Woman, who managed to be less of a stereotype than Apache Chief. Then again, most sports franchise "Indian" mascots are less of a stereotype than Apache Chief. Impala from South Africa boasted the speed and agility of his namesake. Germany contributed the Wild Huntsman, a heavily-armed refugee from Wagnerian opera. Little Mermaid from Denmark had ties with Atlantis. Canada had Centrix; Indonesia, Cascade; Israel, Seraph; Russia, Tundra, and France, Fleur-de-Lis and Chrysalis.
Bushmaster from Venezuela had reptilian powers; he was eventually killed, most likely because "reptilian powers" don't provide a crimefighter with much of an edge against villains. Actually, quite a few of these characters have been killed.
Although Super Friends has never been considered part of official DC Universe canon, the Guardians finally appeared canonically in Crisis on Infinite Earths #12 and thereafter began to battle crime in DC's official adventures. They were given an elaborate history which retroactively tied them in with an organization called the Dome and DC's little-known ethnic heroes of the 1960s and 1970s, Knight and Squire (England), Gaucho (Argentina), the Legionaire (Italy), and Wingman (Sweden). Shortly after being introduced into DC continuity, most of the Guardians were killed.
These heroes did not always fall into the obvious ethnic stereotypes of their televised predecessors. But they rarely have been developed as characters; many have only appeared once or twice. Icemaiden and Fire went on to become Ice and Fire of the Justice League; others have been revived more recently as members of DC's Ultramarines, yet another attempt to create a team of international superheroes. Without greater understanding or experience of those nations and/or ethnic cultures-- and, hey, just possibly, the involvement of artists and writers from somewhere other than Caucasian North America-- they are likely to remain, for the most part, super-stereotypes.
Thanks to Gamaliel for information concerning Ice and Fire.
Pseudo_Intellectual argues that Tuatara's ability to see into time could be related to his namesake's light-sensitive third eye. Possibly, but I cannot imagine they would have named him that if he had been an American hero.