To install Kindle for PC on your Linux system, make sure you've got WINE installed. Most major distributions (Ubuntu, Fedora, openSUSE) offer WINE in one of their repositories, or have versions custom-made for them.
With WINE installed, download the Kindle for PC installer, then double-click on the .exe file you downloaded. WINE will pick it up and install it in its virtual C:/ drive. The one issue you'll encounter is that Kindle will automatically start up and ask you for your username and password, even though you can't see the fields; you can quit the app and fix that, or just type in your username, hit Tab, then your password, and then quit.
Head to your system's menu and then to the WINE folder, and hit "Configure WINE"—if you don't see it, just run
winecfgfrom a terminal or Alt+F2 prompt. In the "Applications" tab, hit "Add Application," then navigate to Kindle for PC in your virtual Windows drive, which is at C:\Program Files\Amazon\Kindle for PC\KindleForPC.exe by default. Hit OK, select KindleForPC.exe back in the Wine configuration window, and change the "Windows Version" drop-down at the bottom to Windows 98. Head back to your WINE menu, run Kindle for PC, and now all your controls and buttons should be showing up correctly.
You can now send books and sample chapters from Amazon's Kindle store to your PC. If you've already installed Kindle for PC in a Windows machine and tied it to your account, be sure to send your Kindle items to your Linux-based reader (which was "Kindle for PC 2" in my case).