Summary: Roaches and filth. Roaches and filth. Go together like, well, I don't know an adequate rhyme, but who cares? Cockroaches are yucky!
Cockroaches are always associated with filth. Why aren't our feelings about ants and spiders and beetles the same? Well, for one reason, cockroaches are known to carry Salmonella and E.coli bacterial pathogens and the other aforementioned little fellows don't.
Put a cockroach under a magnifying glass and you'll see they are covered with hairs and spines. All these things hanging off their bodies pick up bad stuff as they make their way through drains and sewers. If they have just emerged from one of these sewer lines and have somehow found their way into a kitchen. you've got instant contamination. We often hear about outbreaks of food borne diseases and many such outbreaks can be traced back to cockroach infestations.
The lowly cockroach also does a great job of irritating people with allergies. In fact, it is now commonly belived that people can have asthma from exposure to cockroaches. As cockroaches shed their outer skin during molting which occurs six to twelve times during their lifetime, the molted skin decomposes into tiny flakes. Air currents pick up the flakes and they float in the air around us. Eventually, with enough of these flakes in the air, we will breath them into our lungs. Put enough cockroaches in a room and you'll have an appreciable amount of cockroach stuff floating in the air you are breathing. People with allergies are quickly affected. This is an especially well known problem in low income housing where incidences of childhood asthma cases far exceed other socio-economic neighborhoods.
So, there is good reason to strive to live in a cockroach-free environment. Ignoring an existing cockroach population can bring exposure to unhealthy pathogens and potential respiratory problems.