Most of the time, this really is not a big deal. Small pieces of snow or ice, or a pile of powder may fall and get you a little wet. When there are extreme temperatures, though, you really need to worry. Just listening to (and feeling!) my house shake as piles of snow slide off of my metal roof are enough to drive that thought home. You can really get hurt any time that you have really cold temperatures where you can get large chunks of ice or icycles or any time that it is warm enough that the snow turns to slush before falling on you.
Look ahead. See if you can spot anything that has fallen recently or looks like it may fall in the near future. There will usually (but not always) be small pieces of ice and patches of snow that fall in the minutes leading up to a large shake-up. Try to run around anything that might look dangerous, if possible, so that you don’t have to pass directly underneath. If you do have to run underneath, then at the slightest hint of anything moving be sure to cover your head. A cut arm is always worth the trade off when compared to a cut shoulder or a concussion.
That being said, you probably do not really have anything to worry about. This is one of those things that you want to be mindful of, but most of the time when something falls out of a tree you might not be happy if it hits you but you will rarely be hurt seriously. It never hurts to be cautious, though, and to at least be aware of the dangers.