At the LA Auto Show's Design Challenge last year (yeah, we're late on this one!), Mercedes unveiled their eco-friendly version of a woodie: Mercedes-Benz RECY.
Sadly, it's only a drawing - would've been neat to see it in real life: http://www.laautoshow.com/show/DesignLA/DesignChallenge/DesignLAAcura/tabid/95/Default.aspx | Other Entries - Thanks David R!
I can't imagine real wood being very good for collisions. Imagine insult to injury when you get whiplash AND splinters. :P