I have the same problem on my EeePC 701 (chromium-12.0.742.91-1.fc14.i686) for some time now. It crashes shortly after the Chromium window appears. However, the x86_64 build works fine on my desktop and laptop.
I verified the integrity of all the installed packages on my EeePC using rpm so I am sure that Chromium is the problem. I since gave up up Chromium and use Chrome instead.
However, I just built a F14 i686 live USB and Chromium is broken on that too! I can't use Chrome because Chrome's installation does something funky and causes the the live image build process to fail.
In both cases, SELinux is disabled.