Cleaning a cage depends on the material you bought and just how you want to clean it.
If you bought any material other than 'Smooth' then a good scrub while in the shower is sufficient for daily activities. However, you may notice some buildup on the surface that scrubbing does not remove. There's a small amount of porosity in the non-smooth materials due to the nature of 3D printing. For a deep cleaning of those parts, the cage is okay to put in the dishwasher with the caveat that the harsher cleaning environment will cause the dye color to fade faster. The temperature of the water in the dishwasher is well below the melting point of the Nylon the cages are printed with. Secondly, I've had customers tell me they've had great success in cleaning their cage using denture cleaning tablets in a glass of water on the nightstand overnight just as it would be done for dentures. This is a more gentle cleaning, but still gets into the nooks and crannies.
If you did buy the smooth material, then just a quick scrub with soap and water on a washcloth or loofah while in the shower should suffice as it does not have the same level of porosity of the processed plastic.