Hi Adam,
I recently experienced a rankings drop after P3, and here's some of the things I did to correct the problems. Before proceeding, I know that everyone says that Google must re-run Penguin to see any changes. However, from my experience, that depends on the way Penguin affected you.
If P3 penalized you by completely removing your site, then you're probably not going to see any recovery until Google re-runs Penguin. On the other hand, if you experienced a rankings demotions, as we both did, then you can take steps to fix it and will see results without Google re-running Penguin.
Obviously, this is an area of HUGE speculation. I know it seems like Google has stated that a Penguin penalty requires that Google re-run Penguin to see that penalty lifted. However, the Penguin algorithm is just one part of Google's algo, so it only accounts for a portion of what Google considers when deciding where to rank your site for a given keyword.
Further, I'm not focused on a P3 penalty, but rather a demotion. A demotion does not necessarily mean a penalty, but instead could indicate (1) links to your site were de-valued, thereby hurting your rankings, or (2) your competition cleaned up their link profiles and were rewarded, thereby allowing them to outrank you.
Regardless, here's what I did and I hope this helps you.
1. I used CognitiveSEO to go through my link profile. There are other similar tools out there, but I like this one.
2. After going through my link profile, I uploaded a disavow file to GWT. If previously disavowed links are still showing in your GWT, don't worry. They will still show in GWT, but will be marked as "no-follow."
3. Carefully go through your entire site. Ask yourself, "Is this a good user experience?" Be honest with yourself on this point. Is the content good/useful/informative? Is the site nicely laid out? Does the site inspire trust? In short, make it a great site.
4. Internal linking - huge area of opportunity. I would say carefully go through each page and ask yourself how to improve the user experience by internal linking.
5. Promote via social media. Also, carefully study your competition to understand how they are promoting their sites.
I know this may sound cheesy, but I really believe that Rand is correct when he tells us to "be awesome." If you make a great website and promote it, people will not only come to the site, but also come back to the site. Repeat business is everything.
Although I'm certain I experienced the same panic/stress/anxiety you did when you saw your rankings drop, I realized there was nothing I could except really focus on making a better site: content, U/X, conversion driven design, download speeds, social media, and a link profile that I wouldn't be ashamed of. I believe that if you do those things, you'll see an improvement in rankings.
I hope that helps.