Being White, I have the dubious privilege to be able to ignore race in my roleplay gaming and my fantasy fiction. It's a dubious privilege because it's one that is impossible to ever fully decline. That's not to say "poor white me boo hoo"—rather, the only moral response is to decline the privilege at every opportunity. The pervasiveness of White privilege is such that I can never catch every instance, and when I do I won't always know what I can do to reject it. The key is staying aware of the taint that filters my culture, looking for the chance to resist, and learning more about the reality that is discarded by those filters.
On that last point, some edifying links.
Pam Noles' essay "shame" is the personal story of a young girl couldn't find herself in her beloved fantasy books, her elation at discovering that Ursula K. Le Guin's character Ged in A Wizard of Earthsea is brown, and the shameful Whitewashing of the book and its racial message in the Hollywood adaptation of the book.
bankuei over at Deeper in the Game writes about the perpetuation of the white assumption in fantasy gaming by publishers and players. He particularly notes the inanity of a genre that has room for elves, wizards, half-dragon vampires, and lighting–throwing god-child alchemists who can spontaneously grow wings, but doesn't have room for any colour of human except White.
Monte Cook takes on the twin themes of race and gender in D&D art. In a genre that is all about imagining a diverse palette of possibilities, it is particularly odd that every Strapping Young Swordslinger produced by publishers is as White as bleached cotton. (And male to boot.) Monte gives the example of Regdar (an iconic character in 3rd Edition D&D): he was shoe-horned into the books at the last minute by a marketing team who assumed their target audience was male and White and who feared alienating their "core" market of male White gamers if they didn't have a dominantly-raced and -gendered character for the game's launch. Of course, that's not how it was understood at the time, but that's how racism and sexism works: "it's not biased, that's just how the world is". That attitude keeps the world seeming that way. That the design team pushed strongly for diverse art that didn't include the ever present White Male Fighter is great, despite the sabotage.
And finally, keeping this bingo card handy when engaging with race issues is probably a good way to red-flag all the ways in which we've been conditioned to perpetuate and protect White privilege. There were more than a few squares to which my reaction was to say "but that's justified!", only to realise that it was a perfect example of how otherwise good-intentioned people like me participate in the maintenance of racial imbalance.
How do I apply this to my gaming? To be honest, I don't. I'm still trying to figure out how to disrupt the White assumption in my own gaming without it being a naïve effort that ends up backfiring. I've tried playing a brown-skinned man before, but I didn't know what to do with that character detail. Playing it up would have been as bad as Hollywood's magical negro. As it was, it just sat on the character sheet and there was never a moment in the game where it entered into the narrative as a "not a big deal" detail.
As a GM I'm responsible for portraying entire cultures and worlds, and it's hard to overturn the "everyone is white" default without either being ham-fisted about it or Orientalising a culture. One way of overturning the invisibility of Whiteness (part of how it establishes itself as the default) that I've considered is just to describe the skin colour of all my characters regardless of whether they are the invisible White or a marked Other. The problem there is how to describe White characters then: do I just say White? What about actual white skin that a moon elf has? The White race isn't even homogeneous, since it's a modern construction for political and power reasons: real White skin colours range from pale pink, to tan, to olive, to yellow, and more I'm sure I'm missing.
What do you think about portrayals of race in your shared fiction?