This is true, but depending on the OP's version of IOS, this actually might not work. I recently discovered, much to my dismay, that IOS 7 doesn't actually have those sharing options. I discovered this while trying to guide a friend in the best way to save his recordings elsewhere, rather than having to transfer them via Itunes. After doing some digging, since he also required a free recording app that was also compatible with IOS 7, I found an app called Rev Voice Recorder. It has dictation features that most folks probably won't need, but they're optional, and, from what my friend told me, the rest of the app works well and is very accessible.

I say all of this in case this info is useful to someone, since not everyone is fortunate enough to be able to get the latest and greatest phones and software all the time. And, regardless, this Rev Recorder should work on any version of IOS. But if you do have a later version that supports sharing to Dropbox and other apps, then there's no reason why the built-in voice memos app shouldn't do the job just as well.