The safest cleaner for cedar is clean water applied in many light applications. You don't want the water to soak deeply into the wood, just rinse the surface.
Cedar chests must be dusted regularly, but be careful about using typical dusting sprays. Be sure they don't contain silicone which can build up on your chest. You can also vacuum your chest with a brush attachment to get all the dust out of any crevices in the design.