Usually, I wind up using a bit of it at its normal consistency to cook a meal that would call for yogurt or sour cream, and then follow this process to thicken it up. My end result is somewhere between thick Greek-style yogurt and commercial cream cheese, and it works great as either a fluffly spread or as the base of parfait (throw in some crunchy granola bits, dried fruit, bits of fresh fruit, nuts, a drizzle of honey and this is an awesome breakfast or dessert.
I've seen instructions for this that call for cheesecloth and hanging it from the kitchen faucet and all sorts of convolutions like that but that sounds just fussy enough for me to rarely bother (my sink gets heavy use - food stuffs hanging out in there would be so beyond annoying to me).
So here's my lazy girl's method of thickening plain yogurt - get one of those middling sized mesh strainers (mine's 6" across) with a lip of some sort on it so it will rest on the edge of a bowl. The bowl should be deep enough that the strainer allows for at least a couple inches at the bottom - you want to be sure that as the whey drips into the bowl, the strainer doesn't wind up sitting in the whey.
Now, instead of cheesecloth, jus slip a basket style coffee filter into the strainer. Dump the container of plain yogurt into the coffee filter and... that's it. Cover it (a clean dish cloth is fine) and pop it into the fridge.
In about 4 hours, you'll have good thick Greek yogurt. Another 4 hours, and it will be as thick as spreadable cream cheese.
The coffee filter makes it extremely easy to turn it into a container for storage in the refrigerator - the filter can be picked up (carefully so the contents don't fall out), held over the container and one little shake plops the whole thing into the container with very little still left on the filter to be scraped off.
And then, there's the whey - the leftover liquid that has drained into the bowl is a wonderful little bonus - and reason enough to get your 'greek yogurt' this way rather than buying it at much higher cost already thickened.
Pour the liquid into a glass jar (this is a clean spaghetti sauce jar), and keep it stowed in the refrigerator - it'll keep for up to six months at least, although mine never lasts that long.
Whey is an acidic pale yellow liquid that is perfect for adding to dried beans when they're soaking to aid with digestibility, to soak whole grains for the same reason, as a healthy addition to soups or smoothies. You can even stir a bit of it back into the thickened yogurt if you suddenly have an urge to eat it at its thinner consistency. I add a couple splashes to whatever I'm making and by the time it's gone, so is my thickened yogurt, so I do all this again.
All of this takes about 10 minutes actual work total, including washing the dishes used, and yields great returns in terms of healthy refrigerator ingredients and in cost savings. Can't beat that!