Hannah Druckenmiller and I wrote a paper on a new method we developed that allows you to do cross-sectional regressions while eliminating a lot (or all) omitted variables bias. We are really excited about it and it seems to be performing well in a bunch of cases.
Since people seem to like watching TV more than reading math, here's a recent talk I gave explaining the method to the All New Zealand Economics Series. (Since we don't want to discriminate against the math-reading-folks-who-don't-want-to-read-the-whole-paper, we'll also post a blog post explaining the basics...)
If you want to try it out, code is here.