We live in a major metropolitan area, and I recently replaced our garbage disposer. While it was off I took a look at the pipes under the sink and they were just caked with black rotting gunk inside them. I put replaced them all with new ones, and since then our cockroach problem seems to have gone away completely. I believe all that rotting crud inside the pipes was a full time food source for the bugs. Used to be if you turned on the kitchen lights at 2am you would see a scurry of activity on the counters. ApparantlyApparently, now that the food source is gone, they left. Turn on the lights at 2am these days and there are exactly ZERO roaches.
Now, we use the disposer for cleaning the scraps off the plates, but that's about it. If there's a large quantity of something that needs to be thrown away, we'll put it in a two Walmart bags and store it in the freezer till trash day.