You can do it sans machine, but you are limited to small batches, and it tends to form crystals. You have two choices, bags or freezer container. Bags tends to make less crystallized product, and is a bit faster.
Get a large 1 gallon ziploc bag, lots of ice, a ~2 cups salt, a small, 1 quart bag and the mix to be turned into ice cream (or frozen yogurt). Chill the mix in the fridge beforehand. This helps the mix cool more quickly and evenly, meaning smaller crystals, meaning creamier product.
Place the cream mix in the small bag, and leave a bit of air. Make sure it is sealed well, you may want to double bag it. Put the small bag in the big bag.
Fill the big bag about half way with ice and add a few tablespoons of salt. Seal the big bag and start mushing the small bag from outside the big bag, moving it around and getting it in contact with the ice. You probably want gloves or to wrap it in a towel: it gets really cold. After the ice melts down, add more ice and salt, keep squishing until the stuff in the bag gets hard. You can toss it on the floor and squish it with your feet while watching TV or reading. Put it in the freezer for a few hours to condition.
This isn't quite as nice as a machine. To compensate for the extra crystallization, you can add more fat and sugar, since those stop large crystals from forming. With frozen yogurt, I don't think there is much you can do.
This is easier, but makes lower quality product. Simply put your mix in a large container (leave at least half empty), and place in the freezer. Take it out every 10 minutes and shake vigorously for a few seconds. Repeat until it won't move anymore. Use a fork to fluff the mix.