The most commonly recommended cleaner for grinders (including burr grinders) is to run a batch of minute rice through the grinder. The rice will help to pull the oils and other trash out of the system. I usually disassemble my grinder and brush it out before the rice is used but if you clean with rice often enough that may not be necessary. Minute rice is softer than regular rice so many people (myself included) feel that it is less likely to dull the burrs of a grinder. A small amount of residual rice dust in the grinder after cleaning should not add any noticeable flavor to the coffee.
One manufacturer of a commercial grinder cleaner claims that rice can cause issues in a grinder so another option is to use a product specifically designed for cleaning coffee grinders.