The kindle device is capable of reading non-DRM .mobi files. The Kindle app is not.
Your best, easiest way to read these books is the Stanza app. You can import them using File Sharing in iTunes. That's not drag and drop. You need to connect the device, go to the Apps tab and use the File Sharing stuff at the bottom.
If you have a Mac, you can download the OS X version of Stanza and sync the files that way.
You can also look into converting the books to ePub, which will give you other reader options, including iBooks. Calibre is the most commonly recommended app for this. It is available for both Windows and Mac computers. Again, if you have a Mac, Stanza for OS X will convert many formats.
For those that are wondering, yes you can get DRM free .mobi files from several sources, legal ones even. Baen.com offers this type of download and I have several, though I went back and re-downloaded them as ePubs as soon as they offered that format.