My favorite app is the Ambling Audiobook Player. It'll work fine with your non-DRMd MP3 files. Just add the MP3 files to iTunes, then you can access those files through the app.
There is a free,
Lite version, and Pro version with more features that's maybe $7.99?
Alternately, you can listen to the books just fine via the Music app on your iPhone. Click the More... button at the bottom right of the main screen to see the audiobooks option.
FYI, for both options, your MP3 files in iTunes must have Media Kind set as "Audiobook." Highlight the files, right click and choose "get info," click on the options tab and make sure Media Kind is set as "Audiobook." Also make sure that "Remember Playback Position" is checked.