This replaces PC "race" with PC Background.
At PC creation, each players chooses or randomly generates a Background for their PC. Backgrounds are a two- to four-word phrase that describes (i) the basic ethnic/social/cultural origins of the characters and (ii) the character's subspecies.*
So, like, if we're talking Dark Sun, you'd have an Athasian Elf Background. The subspecies component is digital: the PC is an Elf or a Human or a Half-Elf or it's whatever else but it's always just a subspecies. The cultural component of the PC's Background, however, permits a great deal of flexibility. "Athasian" can be reduced to various granules like Silt Sea Elf, or Tyrian Elf or Tyrian Slum-Elf, which is really just a Tyrian Elf with a particular sort of upbringing, etc, etc.
HOW IT WORKS
Option One (mechanics-free):
Backgrounds connote some basic in-diegesis facts about the PC and you can leave it at that: the choice of PC origin and subspecies in this instance would be entirely a matter of preference, a choice free of mechanical considerations.
Option Two (Background Traits):
I find that the liberating aspect of Option One does not compensate for the lost interesting situations/tactical options created by the various advantages normally associated with a given D&D race. So:
New Rule: The PC's Background is also a Trait: the PC's player can try to used the PC's Background to try to earn a bonus to a laterally-related non-attack roll.
New Rule: Backgrounds also permit access to special, Background-specific Traits. As with other Traits, anyone should feel free to spitball Background-specific Traits, but, as with other Traits, all Traits are subject to GM approval and, once approved, can be used by any (N)PC with a related Background. Players suggesting Background-specific Traits must supply a reason why the Trait ought to be available to the Background.
Background-Specific Traits are special and, unlike other Traits, grant either (a) a minor bonus to a skill roll (and by minor I mean: to the smallest significant increment possible, depending on the system currently in place - likely a one point or 5% bonus) or (b) access to a spell from the first level B/X Magic User or Cleric list usable 1/day.
EXAMPLES OF BACKGROUND-SPECIFIC TRAITS
Halflings are small and get a bonus to sneaking
Tieflings are infernal and their nature permits them to cast Darkness 1/day as a first level Magic User
Dwarves get a bonus to detecting doors hidden in stone
Elves in Dark Sun gain a bonus to surprise while in the wastes
More exotic Backgrounds should still stick to Traits that work as in (a) or (b) above, though it may be necessary to slightly hack or re-skin the spell or ability in question. Like:
Midgard Gearforged can repair minor damage by spending 4 uninterrupted hours tinkering on themselves or another construct - restores 2-7 hp as cure light wounds 1/day
Lightning Genasi get a single static zap that deals 2-7 damage and functions as a Magic Missile cast by a first level Magic User
PUTTING IT ALL TOGETHER
A PC with the Tyrian Elf Background might get access to these two Traits:
1. "Standard" Trait: The PC's Background as a Tyrian Elf can be leveraged for a minor bonus to non-attack rolls specific to Tyr
2. Background-specific Trait: Elves in Dark Sun gain a bonus to surprise while in the wastes
"Races" in D&D (a) connote a constellation of genre referents/setting-specific elements and (b) denote the mechanical translation of that constellation. Backgrounds explicitly mimic this structure in a way that I hope is perhaps a bit more explicit and easy to mess around with/retrofit to a specific setting.
OD&D's race mechanics seem unnecessarily convoluted and lean heavily on Appendix N-associated tropes. Backgrounds aim to offer a possible, streamlined alternative.
*I like subspecies over "race" or species. I sort of assume that any playable "race" in my campaign that is at least mildly humanoid and of the same assumed genus can interbreed, though we never really get into that kind of thing.
pictures: top by Andy Weber bottom from the internet (original source couldn't be located)