One can build a compost pile and hurry the process along with maintenance of correct moisture, correct ratios of nitrogen (green stuff) and carbon (brown stuff) and oxygen and with a turning schedule timed to keep the most effective type of bacteria working and reproducing at maximal efficiency. This is a good way to get wonderful humus for the your soil and exercise in the meanwhile. My compost pile is my gym. I haul hundreds of big bags of leaves, pine straw, and grass clippings from neighbors' yards every year. I turn it (spread it out and rebuild it once it cools down). I add my household vegetable wastes to it everyday. In the spring I dig it into the ground before adding new plants to the yard. In the fall I use it for mulch. When the microbes are most active it is so hot that it can burn your hand in the middle. The mass of material constantly shrinks as the composting takes place. In January it is typically about 1/4 the size it was in October.
One can leave Mother Nature to her own devices and just leave the organic stuff laying around instead of being too fastidious about raking up leaves and picking up the grass clippings.
One can pack it up and send it to the landfill or if the local jurisdiction is wise send it to the government compost pile.
One can rely on aerobic bacteria and fungus to break the stuff down (quicker and smells better) or allow the anaerobic guys to do the job over a longer period of time and put up with the stink.
One can lay organic material on the ground and allow the worms and friends to digest it, pulling it down into the soil in the putting the annual garden to bed:process .
One can bury the organic material in small spots or trenches and feed the worms that way.
One can even bring the worms indoors and set up special bins allowing worms to crawl up through bedding material to kitchen wastes laid on top where they eat, digest (compost) and reproduce. There is even a book and multiple web pages dedicated to the subject - “Worms Eat my Garbage” by Mary Appelhof. Her page is at http://www.wormwoman.com
But anyway you look at it, compost happens. Dust to dust …
Many links on composting:
Use of composted organic materials:
household (backyard) composting: