Powder skiing is all about making little adjustments. As Castle Dave has noted, you can adjust your gear to the point where groomed snow technique will work just fine. With your normal gear, the natural adjustment to avoid going over the handlebars is to sit back. When you sit back, you end up burning your quads quickly. This works, but not very well and not for very long.
So let's add the classic adjustments that Dave has hinted at to the earlier advice to come with a starter list of powder tips:
Move your feet a little closer together than normal (less than 6 inches apart, but not locked together)
Make your weight distribution more towards 50-50 than normal (a lot more weight on the outside ski)
Make your turns a little narrower (less across the trail)
Make your speed a little faster
In less than a foot of snow, this is usually enough. But over a foot, that "submarining" feeling needs one more little adjustment. If you lift your toes up (close the ankle joint), you can get your skis to "plane" against the snow the same as leaning back, but with a lot more control and a lot less quad burn. The exact amount of plane that you need will vary with the snow depth and consistency, your gear and your speed. This is not easy.
Since it's hard to get the right amount of plane dialed in right away. I help rookies get started by starting them off in a traverse to help them find a straight line speed they are comfortable with while playing with the "plane angle". When speed gets too high or submarining starts to happen, turn uphill to a stop and reset. When they gets comfortable, try bouncing in the snow while traversing. When that gets comfortable try turning down the hill at the top of a bounce. This is where you apply the advice that Dude and TPJ have provided. Eventually, the bounce happens naturally as the rebound from one turn to the next and ouila you are skiing powder.
One more tip I give to newborn powder hounds is to make your turns in slow motion. When you are in the snow, the snow provides resistance to your movements. Quick movements get punished. Patient movements overcome the extra resistance.
Remember that the amount for all these adjustments will vary depending on lots of factors and there are no friends on a powder day.