I've read of the frustrations of many fellow users of Audible over the lack of ability to tell the Audible app to use a removable SDHC card if a device has a slot. Well I just downloaded the newest version 1.4.5 Build 504(of course from Google Play). It works perfectly. It is not ideal but I find their solution very workable. All you need to is change the setting for downloaded files in the "SETTINGS" section to point to whatever storage location you want to use. It can be the internal or either removable card slot (you know in case your device has a dock with a second card slot like the Asus Transformer family).
Today I was able to free up ~20GB of on device memory by pointing to the micro SD card I keep in my TF300t. I let Audible do a refresh of my library and viola it found the files I already had on the card. Of course you either properly create the "Audible" directory on the card as a place to save the files. There are instructions around the web on that but if anyone needs them I'll root around for them (hey it's about 2:30am and I need to get back to sleep but was excited this worked..
When you point the app to your card it does not move the files for you so you need to either copy the existing Audible directory or manually create then copy the files over to the card. I did not spend enough time to check but I assume you need to leave the regular Audible directory on the internal memory but then again maybe not. I left well enough alone for tonight and just deleted the files in the default directory Audible uses for it's files (config as well as audiobook) leaving the other files and subdir's alone. Will test more when I have time later this week.
I finally feel like I can use my tablet for one of the primary uses I have for it. Of course this also should work fine on a phone with an SDHC slot. Only prob I have now...I will soon need a 64GB card in order to keep all my audiobooks handy. I like to have them all when I travel because I never know what I might be in the mood for at that particular microsecond