How does one treat a cockroach bite?

Crickets of most species have the horsepower and strong mandibles to inflict a bite. They generally don't have the strength to get through the skin of the hands, but softer skin is prime as a target. They apparently feast on blood, as do so many insects. While I can't find any literature on crickets carrying any specific pathogens (which is good), blood feeders can do this (which is less good), so cleanliness is the way to go here. Clean the bite thoroughly in warm, soapy water. Be thorough and take your time. Dry the wound with a sterile dressing (or let it air-dry). Apply either Betadine, hydrogen peroxide, or (my favorite), a combination of both. Bandage the wound (a Band-aid should do), and watch it.

If it heals (which I expect), wonderful! If, however, it remains red or gets redder, starts feeling hot to the touch, or otherwise acts infected, head for the doc, for antibiotics.