I am shocked at the enthusiasm of folks for playing a book on a Roku. The idea of running a television to listen to a book or watching a book on a tv does not appeal to me at all. Audio books seem to pair most naturally with an MP3 player or a stereo -- something portable.
That said, if you have audio books in MP3 format, I'd recommend getting a Roku with a usb port so you can play the books whether your internet is slow or disconnected. I like the Nowhere USB channel for this mostly because it can play one file after another without intervention. This would be great for books that come in multiple parts or chapters -- name them correctly and you are good to go.
I'd get one with composite/analog audio out so you can plug that into an aux audio in on your HTIB or even amplified speakers.
You can get a refurbished Roku 2 XS for ~$50 these days and that is probably the best option. It has the usb port, composite out, and both wired and wireless ethernet. You can use it with no ethernet at all if you are streaming audio books off a thumb drive with Nowhere USB. As a bonus, you can play Angry Birds with its motion remote control.
If you are new to the Roku, take a look at
this. I have six of them and everything I have learned is posted there.
Here is a link to a post on some USB channels I have looked at.
Happy streaming!