I have the same issue with voice quality and cutting out on my Verizon Moto X. I ordered mine online on September 1 in black. If other people are not experiencing this, I wonder if there was a bad batch at the online warehouse or if there is some other common thread that is causing this. I am going to try to swap mine out at a store and see if it does the same thing.
I can reproduce it on the phone using the built in mic. It does not appear as bad when using the speakerphone or Bluetooth, which makes me wonder about noise cancellation. I have confirmed that it happens whether I have a case or no case on the phone. I looked to see if there is lint or anything in the mic, but it is clear. I had people asking me to repeat myself from day 1 in using the phone. It almost seems like a full duplex issue. If their phone is transmitting anything, then my mic is muted partially. I have also had people not be able to hear me when they are not saying anything, but maybe their phone was sending some background noise that I did not detect. I have not asked anyone to mute their mic and see if they can hear me better, but I may try that next to see if it helps as a test.
People keep talking over me because they can't hear me. They also tell me I am cutting out. Sometimes when I ask a question, I get silence because they did not hear it.
I talk on the phone a lot, so if this is an issue on a replacement phone, I am going to have to switch to a different model. I really like the Moto X, but I need to be able to use it as a phone and have people hear me.