I've ridden sections of 66 from north of Prescott,AZ (west of Flagstaff) all the way to Adelanto, CA, where I peeled off the route, which, I think, turns into massive interstate.
There is much fantastic cycling here. You'll need to study the maps and see where you'll need to leave the old route, but Ashfork to Kingman, AZ, is great. Then you'll want to the climb over to Oatman, which is one of the best stretches of road I had the privalege to ride on my cross country trek last year. In California, you'll play tag with Interstate 40, but when you leave it, as in the drop to Amboy, you'll have a blast. One section to keep in mind: From Ludlow to Barstow, Route 66 parallels Hwy 40, which seems nice at first, but the pavement is unspeakably bad--like, OH MY GOD I CANT BELIEVE PAVEMENT CAN GET THIS BAD! The last two times I've ridden this section (once east bound, once west)I defied the law and used the shoulder of Hwy 40--wide, smooth, quick. You can get off at Newberry Springs and check out the classic Baghdad Cafe. From there, Route 66 has good to excellent pavement.
I crossed the Mojave in early November, which was great. Winter is generally excellent, although you could face some occasionally fierce headwinds as you go west. If you are unlucky, you could hit winds that are almost impossible to ride against. The desert can be a harsh mistress. Fall, however, is probably the best time to go east to west. Personally, I would never do it in winter--too freakin' hot!
Go for it, and have a blast.
Cheers,
Scott