I've searched high and low for the answer to this. Most of the tutorials on other threads say, "It's user error," when, in reality, it's a software problem. I've tried different cases, different grips, changing the volume, different talking angles, and turned off the noise cancellation feature. Nothing fixes this, and weird as it sounds, it always happened when I held the phone up to my left ear.
I spoke to an associate at my Apple Store who said that the problem was just coming to light, on their end. He couldn't provide any technical explanation, but apparently the issue is with those of us who have never done a clean installation on our new iPhone. I've been restoring backups to my new iPhone since I started with my iPhone 3GS.
Last week, I finally had the time to do a completely clean install on my 5S. For the first time, I can hold the phone with my left hand, hold the phone to my left ear, and nobody on the other end says, "Take your hand off the mic!"
Hope this helps.