I run Linux Mint on my primary desktop. I also have a subscription to Amazon Prime. My music tastes shift around and I slowly drift between the various music services I have access to, as well as my personal collection. Recently I tried the Amazon Music player in Chromium 59 on Linux Mint 18.2. This was my first time attempting to play music from the Amazon player on Linux and it was failing. If I tried to play music that wasn't in my own library, it would simply skip through all the track titles instantly and then pop up an error at the end of the playlist. I attempted to Google the problem and was met with lots of ranting about Amazon streaming products and Linux. I was on the verge of giving up when I got my first clue:
At the end of the address bar was a little icon telling me that a plugin was disabled. Specifically, Chromium (and Chrome) disabled Flash by default as part of the effort to phase out Flash support. The Amazon player uses Flash, and thus was failing to play because the plugin was not loaded. Clicking on the disabled plugin icon produced a small window where I could either enable it temporarily or visit another page and enable it permanently.
Clicking on Manage plugin blocking takes you to a page where you can add URLs to a whitelist for the Flash plugin. I clicked Run all plugins this time and verified that the player worked properly, and then added the url https://music.amazon.com to the whitelist.
Now when I open the Amazon music player, the Flash plugin is automatically loaded and it can stream music properly.