It’s no secret that Amazon’s business strategy involves growing its digital media sales, which it has done by undercutting MP3 music download prices on iTunes. But Amazon’s streaming music strategy, kicked off by its Cloud Player release last year, has left iPhone users out in the cold until now.

Today the company released a version of its Cloud Player for the iPhone and iPod Touch, which will allow Amazon customers to stream or download music stored in their Cloud Drive. They’ll also be able to manage music libraries and create new playlists. Until now, music stored or purchased on Amazon was only available for playback on its website or through mobile devices running Android.

Amazon first rolled out its Cloud Player almost a year ago in a rushed attempt to push out a cloud-based music service before rivals Apple and Google. The Cloud Player is part of Amazon’s Cloud Drive. Users can get 5 gigabytes of storage to store content for free and pay an annual fee if they want additional storage. Songs purchased from Amazon don’t count toward that storage limit. It’s possible that other kinds of media could be added later, but for now the service is limited to music.

The move to add a Cloud Player app to iOS devices mimics Amazon’s efforts with its Kindle store and e-books. You can’t make a purchase through these apps, but you can manage your digital purchases. The Cloud Player app should have launched along with the service itself almost a year ago. It’s delay may have hurt any real chance of Amazon challenging iTunes on the music front, but at least now there’s a comparable alternative to Apple on the iPhone.

The app, which is available on iTunes today, is free to download, as is Amazon’s Cloud Drive service.