This is a great question; I have an interesting book that gives an answer to it: What Einstein Told His Cook, from Robert L. Wolke.

Chemists have identified from eight hundred to fifteen hundred different chemicals in coffee. As you can imagine, removing the 1 or 2 percent of caffeine without ruining the flavor balance of all the others is not small trick. Caffeine is removed from the green coffee beans before they are roasted. First they are steamed, which brings most of the caffeine up to the surface, and then the caffeine is dissolved out by a solvent (ethyl acetate, for instance). To be called decaffeinated, a coffee must have more than 97 percent of its caffeine removed. Ethyl acetate is a chemical found in fruits, and, indeed, in coffee itself, then it can be said to be natural. Methylene cloride is also used as a solvent to remove caffeine, but the FDA limits its amount in the finished product to ten parts per million; his is because in the 1980s, methylene chloride came under fire as a carcinogen.

Another indirect method to decaffeinate coffee is the water method. Here the caffeine together with many desirable flavor and aroma components are first extracted into hot water. The caffeine is then removed from the water by an organic solvent, and the now caffeine-free water, with all of its original flavor components, is returned to the beans and dried onto them. The solvent never actually touches the beans.

There is also the ingenious Swiss water process, which washes the beans with hot water that is already fully loaded with all possible coffee chemicals except caffeine, so there is no room for anything but caffeine to dissolve into it from the beans.

All methods to decaffeinate coffee are safe, so you can enjoy your decaffeinated coffee and forget about the technology used in the process.