The simplest type of greywater system is what Art Ludwig calls the drain out back.  In other words, you take the pipe from your sink or bathtub and run it to an out-of-the-way location to leach directly into the soil.

We only have running water in one location in the house (the kitchen sink), and our original treatment system soon devolved into a drain out back when I didn't have enough wood chips on hand to replace the mulch.  As long as your outlet isn't close to a stream or to a direct route into the groundwater (such as a sinkhole), this low tech solution really isn't so bad.  Yes, the ground gets a bit swampy there, but the soil microorganisms make short work of nutrients and germs in the water, and by the time the liquid seeps down into the groundwater, it's pure.  In most states, new systems of this type are probably illegal, but even the regulators don't really care as long as your system is grandfathered in and doesn't bother your neighbors.

An alternative extremely low tech solution is what Art Ludwig calls "landscape direct".  We use this technique by setting our wringer washer in the middle of the yard and simply dumping the used wash water into the ground.  A more elegant incarnation is the bathing garden illustrated in Create an Oasis with Greywater, which consists of a raised, cobbled mound on which you shower, underlain by sand to expedite drainage, and surrounded by plants to soak up the water.

The main problem with low tech systems like these is loss of efficiency.  If you live in a very dry climate, every drop of water is precious, so it's worth spending a bit more effort to ensure your greywater ends up feeding plants.  Stay tuned for the rest of the lunchtime series (picking back up Thursday after our regular book club discussion) for slightly higher tech, and more efficient, greywater options.