I think Jenn's rationale is a compelling reason that you would want to choose one direction or the other. If you start in May, I'd probably go west to east. If you start in September, I'd probably go east to west. If you start in June or July, then you're just crazy.
But there are no hard and fast rules. You can cross the Mojave Desert in the middle of the summer if you tolerate heat well and carry gallons of water. I loved the Mojave, especially the area around Amboy. I started in Chicago this August and went east to west. East to West seems more natural, as Route 66 is better known as the route to California, not the route to Chicago.
Wind is highly variable (in most places, but in Wyoming, you can pretty much count on a westerly), so I would not use wind direction to pick my route direction, but I do believe (and my wind studies bore this out) that winds on Route 66 slightly favor a west to east direction. Again, I would use other factors to choose my direction. See the link below for the results of my wind studies on four ACA routes.http://www.crazyguyonabike.com/doc/page/pic/?pic_id=2473525
Route 66 is a great route. You'll love it!