So after discovering that FO3 and FO:NV both work perfectly well using Wine, I set out to see if I could get TTW working in Wine as well.
My system is a Core i3 with a Radeon HD7870, running Mint 17 XFCE and Wine 1.7.31, and installing TTW 2.7.2
The good news is that I was able to play TTW in Linux. The bad news is that (1) The installer didn't work at all, so I had to boot to my Windows partition to run it and create the TTW folder, (2) I had to do a manual install because FOMM crashed too often in Wine, and (3) the game still has major bugs.
After all of that, the game remains almost totally playable. There were just three bugs that I could see so far:
Bush textures look weird, like something messed with their transparency and now they all look like they have fat flat branches.
Seemingly randomly, certain things will disappear all at once: hair, lights, the HUD, the player's arms/weapon, and dialogue. It seems to be affected by which direction the camera points.
Music would cut in and out, especially during dialogue, but sometimes it would come back.
Anyway, I'm not seeking tech support, and I'm giving up on this project, but I just wanted to let you all know it's quite possible. I haven't seen anyone else try it.