Cockroaches are one of the cleanest animals on Earth. Their bodies create incredibly powerful antibiotics that allow them to thrive in sub-par environments. They will not carry harmful pathogens unless they are initially exposed to them (the same can be said for any other animal, including humans; if you get the flu, you're carrying the flu lol)
The only problems I have with roaches is cleaning old, dry bins with no substrate. The dust can get in your nose and make you sneeze, water, and itch a lot. However, the same thing happens to me when I clean mealworms or the lint on the lint-catcher on my dryer. Just put on a breathing mask when doing large cleanings.
Otherwise, roaches are very clean. The only other problem that can arise is when food is left in the enclosure too long and it molds. If this happens, it's your fault for not removing it, not the roaches'.