[Update: I now recommend using these steps to install the Google Play Store (which does not “root” the device and does not cause any permanent changes), then buying the highly rated Listen Audiobook Player (which has up to 3x playback speed with pitch correction and a slider that shows your place in the book and how much is left–while taking playback speed into account) in the Play Store for $0.99. The entire process takes about 15 minutes.]

Amazon makes it impossible to import audiobooks into the Audible app, probably wanting you to buy all your stuff from them and under their control. I’d actually be more willing to use Audible if it let me import the many audiobooks I already own from other sources. Most Fire Phone audio apps are useless for audiobooks since they do not let you browse the audiobook’s files, instead treating the audiobook as a song album and and making a complete mess out of the order of the tracks. Another issue with music players is playback speed. I usually like to listen to audiobooks at double speed (and usually more if I am able to fully give the book my attention), but most music players I’ve tried on the Fire Phone do not have a playback speed feature.

I was almost losing hope that I would be able to get a proper audiobook experience out of the Fire Phone, until I happened on the Rocket Player App, which has almost all the required features for an audiobook player:

1. It allows you to browse the files on the phone and keep the proper order of the tracks (while others players mess up the track order). If the track order is still messed up in Rocket Player, use a free and open-source Windows program called Mp3BookHelper (Project Page | Download Link) to rename the tracks (both file names and the Title ID3 tag) sequentially.
2. It has a playback speed setting (after buying the $4 premium version of the app) with pitch correction. The playback speed can only go up to double speed, which is pretty good but I wish it could go up to four.
3. It remembers your place in the book, even after closing the app (provided that you do not use the app to listen to other things, which is quite doable since there are many other apps optimized for music listening).

Steps for importing your own audiobooks on the Fire Phone and playing them using Rocket Player

1. Install the free Rocket Player App, then upgrade it to the $4 premium version.
2. Move your audiobook into a folder on your phone. You can use the USB cable or, if your laptop supports bluetooth, you can use that too, though USB is much faster.
3. If you used the USB cable, unplug it, otherwise the audio player may not be able to see the new files.
4. Tap the “Folders” tab in Rocket Player. Browse to the audiobook folder (but don’t go inside the folder). Tap the folder and hold, until a menu comes up. Press “Add to playlist” and create a new playlist. Now you can go to the “Playlists” tab to find the audiobook and play it.
5. In the Rocket Player settings, you can find the “Playback speed” setting and change it to what you like.