It took me WEEKS to find a solution that ACTUALLY worked, and didn’t involve manually building a nuclear reactor or hand-coding the OS of the space-shuttle from memory. THIS POST is how. (Credit for the solution goes to ubuntuforums user Flanmaster, BTW.)
I tried it under WINE 1.4 with Kindle for PC version 1.10.5 and it actually works. It doesn’t crash, it doesn’t have errors written in gobbledygook, and it actually downloads the books you want to read from your Amazon account. (Login by running the software, and then entering your email + PW and hitting enter.) I should note that my WINE/Kindle is set to “Windows XP” for which version of Windows it’s faking.
Most of the really-complex WINE stuff is over my head, but look- this method just works. And most of the 20+ methods I’ve found (and, sadly, repeatedly tried!) online do NOT. Basically, you just install The Microsoft Visual C++ 2008 SP1 Redistributable Package (x86). The download link on the page (under "free download") goes to the Microsoft website official download page (check the url) so no possible shadiness. While I’m not completely sure *WHAT* it does, I know that it works- I ever installed it AFTER installing Kindle under WINE, and it still worked. 🙂