There are a few things we can be certain of. All we have to work with is the basic headphone plug, meaning the microphone has to use only that to function. The mics seem to function on LP settings, allowing only basic, low-quality communicating or recording. But this may well be enough for phone conversations or some apps that require microphones.
If you have a cheap headset/microphone with a single headphone jack plug, you're welcome to try it, or these will allow you to experiment for the very least amount of money.
These iPhone tailored hedphone/mic combos ($2.79) fit the bill. This Samsung headset should also work and is of a slightly higher quality.
Ok, so now that you've got something plugged in, what can you use it for? We've got a few good leads. People have been able to use sound recording apps like netmemo and Tape-a-Talk, both of which must be side-loaded from Android Market. An app for tuning instruments called Tuner-gString has been found to be functional as well. But where there hasn't been a lot of documented experimentation is with actually making phone calls either through Google Voice with an app like GrooVe IP or Skype. It seems like this would also work well for one of my all-time favorite apps, Soundhound too.
GrooVe IP (+Google Voice account) Skype
Note: Using Skype, at best, would allow only audio communication. If the Amazon version doesn't run, you might have luck with the GetJar version.
Again, big kudos go to grvthang and others for giving us something to experiment with. If you have any luck getting anything to work, be sure to let us know in the comments so we can expand our use of the Kindle Fire! As I said before, we're sure some use of a microphone is possible, but we don't know what the limitations are. We'll be updating this post to include any breakthroughs or dead ends. Thanks for your help!