In Firefox, if you have the Adblock Plus extension installed,
right-click on the "Doodle" image, and Adblock will allow you to block
the image. I chose the option that blocks the sprite on the search page, which has at least blocked the annoying image on new tabs, in new windows, and after re-starting Firefox.
Haven't tried it with Chrome, but would assume that the Adblock Plus beta extension works there as well.