I decided to try another Moto X, this time on Verizon, and it does the same thing. Every other word cuts out. I can't hold a conversation on this phone. The last one was on AT&T. How is no one else having this problem?
Edit: I found some threads about the same issue on Android Central. I feel a little better knowing I'm not all alone, but still confused why some people are having no problems at all.
http://forums.androidcentral.com/moto-x/312820-voice-cutting-out-other-end-call.html
http://forums.androidcentral.com/moto-x/311563-phone-cutting-out.html
Edit: Conversations are fine with a headset. With the speakerphone or using the phone normally too many words are being lost to hold a conversation. My mom said that it sounds like I have a really bad signal, but I have a strong signal and it sounds great if I plug in my headset. Bluetooth in the car has constant popping and cracking. I gave up and returned the second Moto X.