Open Amazon Music through the Alexa app. You can then play it through any of your Echo devices.
Great, I finally found it.
I was looking in Amazon Music and not Amazon music via the Alexa App.
Works great thanks, however this has shown me some major shortcomings of using it this way.
1: You can't scrub thru a track. It's either Play, Pause, Stop and that's that.
2: When it comes to searching for music it's terribly limited, and really just a wild guess, it's mainly suited to YOUR music.
Just going to the music area and onto Artists, which are listed A-Z you can't even select say 'V' to jump to, but scroll down and down and down and down till you get to the 'V' section.
Lets say then you find Vangelis. He will be listed, but none of his tunes, you just touch his name, and every time you do, you get a random one of his tracks.
Pretty poor really for the app to be like this.
As I say, as long as you have added music to your library, then it's much more usable.