Looks like they don't support Roku's just yet. Here is their response.
Hello Robert,
I understand that you would like to access Amazon Music Unlimited on your Roku device.
I would like to share an information that, at this time Amazon Music Unlimited is not compatible on Roku device. I am sorry for the disappointment. Amazon Music Unlimited supports these devices :
-- Supported devices
- Listen on any of your compatible Amazon Music devices:
* Android devices
* iOS devices
* Alexa devices (Amazon Echo, Echo Dot, Amazon Tap)
* Amazon Fire TV devices (Amazon Fire TV and Fire TV Stick)
* Fire tablets
* Computers (via Amazon Music for Web and the Amazon Music app for PC and Mac)
* Third-party speakers and sound systems that support Amazon Music ( Sonos, Bose SoundTouch, Play-Fi, and HEOS devices)
-- Streaming
* Stream on one device at a time.
-- Downloading
* Download music for offline playback on compatible mobile devices (Android, iOS, Fire tablets).
However, I've passed your message to the Amazon Music team for consideration as we make future improvements. Customer feedback like yours really helps us continue to improve our products and provide better service to our customers.
In addition to our large selection, one of the benefits we try very hard to offer our customers is convenience. I'm very sorry for the inconvenience you experienced in this case.
We look forward to a very warm and fruitful association with you!!
We'd appreciate your feedback. Please use the links below to tell us about your experience today.