How about straight replacing 'Race' with 'Culture' and incorporating the Non-Human Cultures into this?
After all Race as a separate aspect hasn't existed in all editions of D&D - remember the Races-as Classes? You could have Cultures including 'Mountain Dwarf' or 'High Elf' alongside 'Human Barbarian' or 'Civilised Man', etc.
. . . that would work reasonably well in my game world.
I currently run a 4E game which allows all the races mechanically, but presents many of them in the game world as members of 7 different "tribes". I have taken pains to make the tribes distinct but generally more human-like in apperance, as I wanted to move away from the over-exotic feel of PC races in 4E as a whole. (In the interests of saying "yes" to players who want those super-exotic characters, I still allow them to play PCs that basically ignore my tribal mappings though . . . and I'd want to continue that if i moved to 5E).
Anyway, I could in theory in 5E go the whole hog and homebrew the actual tribes with full statistics, as Races. I did that in 3E (because e-Tools supported it). 5E looks simple enough as a system that I could do something similar there. 4E was too complex, and tools in DDI don't support that kind of home brew, so the mapping to 4E "build race" was a pragmatic way to do the same.