If you google "running form" or "barefoot running form" you will see that it a pretty heavily discussed topic lately. Here is my two cents.
I had my young kung fu students doing some short fast (they all define fast differently) runs and on the first run down the gym floor they sounded like a hailstorm. I cringed at the thought of bruised heels and sore knees and stopped them.
For the next run I told them we were going to do the same run, except they were to run like ninjas.
I told them I wanted them to run silently. I wanted them to not make a sound.
I told them, "When I say go. . . Run like a ninja!"
They got in their starting positions and narrowed their eyes. They got quiet, there was no more chatter.
Rather than the sound of thunderous hooves I heard the sound on ninja feet. Some of them really got into it and squatted low as they run.
I don't advocate a low squat for running form, but I do recommend running like a ninja.
Run quietly, almost sneakily, to run quietly you must lessen the impact of your feet. It is hard sometimes to think about the position your knees, hips, arms, feet, and upper body should be in. I discovered, however, that for some people, telling them to run quietly corrects a lot of running form issues.
So, try it for a short run (as any changes to running form should be done gradually).
Run like a ninja.