Cockroaches are the blight of many an urban area. They feast on our food and garbage, and quite sturdy against many biological threats, and few things are willing to eat them. Also, they're gross and they smell bad. So, they proliferate, to our great annoyance. Usually, we try to poison or trap them, with some tricks proving more effective than others.
I'm imagining either a robot, or a specifically bred animal, that seeks out and consumes cockroaches for power. If it were a robot, it would need a bacterial digester to turn the roaches into power. (Animals have a digestive system that turns what they eat into ATP and carbohydrate chains that they can burn for energy.) If it were a robot, it would be programmed with roach-like habits, like avoiding light, and tracking pheromones. If it gets too much power, probably it can go plug itself into the wall, and save you a few cents on your electricity bill. An animal version would, if it fed well, attempt to breed, which we clearly want to encourage. Answers.com suggests that a good starting point for a professional cockroach predator would be the gecko, a small lizard with an immense hunger for insects.
I would want to make this cheap enough to drive roaches into near extinction in cities that expressed interest in this. Cockroaches would continue to survive, if they learned to avoid human settlements. (Already cockroaches know to avoid flickering lights, as this means that a human is coming with things like squishy shoes, poisons, and possibly assistant animals like cats and geckos.)