Woolie, I'm in Nashville (Lenox Village). Let's get more people to represent!
inthebiz, Nashville is very un-Mustachian, in structural ways more than how people act. What I mean by structural is that there are barely any sidewalks around the city, unless you're around the center of the city. You can forget about sidewalks in less populated areas, especially Brentwood. Franklin is pretty good with them though. What I'm getting at is that if there aren't a lot of sidewalks (and they wonder why the state is so obese?), everything is car-oriented and it can be downright dangerous sometimes to bike (a plethora of very narrow 2-lane roads). That's something to keep in mind, transportation is a big chunk of people's budgets around here.
All that being said, many of the people aren't too into keeping up with the Joneses. Sure, you have the country stars and McMansions in Brentwood and Forest Hills, but I would wager that many people aren't too in debt. I read an article recently about the opening of the Bentley dealership in Nashville. The general manager said he was optimistic about his prospects because there isn't a large luxury car market in Nashville, even though the wealth compares to some of the coastal areas (healthcare, music and education really do make people rich).
So, you wouldn't have too hard of a time living a Mustachian lifestyle, except for the fact that if you live anywhere outside of the 3 areas you mentioned, you'd most likely have to drive because it is very suburban and roads aren't conducive to biking. Out of those 3 areas, I would prefer, in the following order, West End, Franklin, downtown. West End comes first because it is in the city, yet it's easier to live since there grocery stores and other essentials in the area. Downtown living is very new to Nashville, so you'd have to drive out for daily necessities since there aren't any in downtown. I really like Franklin because it is a compact town for the most part (minus the Cool Springs area), and you could definitely bike around the town. It's historic and not very commercial either, so I think a Mustachian lifestyle would go over well (West End is a bit materialistic).
FYI, avoid the Gulch like the plague (it's very commercial and "trendy" aka $2000 rents right by a railroad track, yay!). It is walkable, but very materialistic and un-Mustachian. I never understood the appeal of that place. If your job does take you downtown, you could consider East Nashville. It's gentrifying, which means it's full of hipsters/younger people rather than 5-Series-driving moms, so there's probably a few Mustachians out there (holler!).
I could go on and on, but I'm out of gas. Pm me if you have some more questions.
And let's get some more Nashville Mustachians to come out of the woodwork!