Adobe flash player not working with Amazon Prime

I had this same problem on Ubuntu 12.04 using Chromium. I fixed it my following the instructions on the adobe help page for installing Hardware Abstraction Layer module:

Prerequisites for protected content playback

For Ubuntu 10.04 or later, ensure that the Hardware Abstraction Layer module is first installed using apt-get. (Watch carefully for “hal” install errors, as a damaged package install can continue to affect video playback.)

sudo apt-get install hal

After the "libhal" (HAL) library install completes, close the browser and clear the Adobe Access directories by executing the following shell commands:

cd ~/.adobe/Flash_Player
rm -rf NativeCache AssetCache APSPrivateData2

Note: If the Hardware Abstraction Layer module is missing, Flash Player still functions. However, it cannot play protected content that requires the Adobe Flash Access DRM (Digital Rights Management) module.

This is probably not working in Google Chrome for Linux because of the limitations of the Pepper Flash as Google mentions on their support page:

Linux users: Adobe Flash Player uses a cross-platform API, Pepper API (PPAPI), to run its plug-in in Chrome. You might notice some of the known issues below specific to Linux Chrome:

[...]

• Viewing Adobe Access (DRM) content is not supported, since Adobe does not support it on Linux.

[...]

I think it's very much likely that:

• Support for the older 11.2 Flash for Linux (using the regular API) has been dropped by Amazon.
• Support for the DRM required in 11.7 in Chrome for Linux does not work because of the aforementioned limitations of it.

This makes Ubuntu fall between two stools in the support for watching DRM-enabled video.

try opening this page. You will see a small box with the title version information. Furthermore, it will run a short flash video as a test and tell you

1) if your flash plugin does function properly

2) the exact version of flash you have installed (good basis for further troubleshooting)

Even though it is no solution directly, it might be helpful for you.