I am having the same issues with my Note 3. Seems like it's an issue with the phone itself and not any of the carriers. I really love the phone and the sound quality with music and video playback is great. This ONLY seems to happen during a phone call. I'm thinking about contacting Samsung to see what kind of a response I get from them. I've only had my Note for 3 days and nothing I do seems to work. Believe me when I say I've tried everything. I hope there is a software fix that they can do and it's not a hardware issue. 
I am having the same issues with my Note 3. Seems like it's an issue with the phone itself and not any of the carriers. I really love the phone and the sound quality with music and video playback is great. This ONLY seems to happen during a phone call. I'm thinking about contacting Samsung to see what kind of a response I get from them. I've only had my Note for 3 days and nothing I do seems to work. Believe me when I say I've tried everything. I hope there is a software fix that they can do and it's not a hardware issue.
First and foremost, if you previously had Verizon or have Verizon then, you know it's NOT Verizon's Network.
Secondly, I rarely make an appearance in these Big Red forums but, I feel that I must because, tragically, my sister is in the same boat as you and your wife. She initially activated her service (after her contract ended with a different carrier which shall remain nameless forevermore) on launch day, Thursday, October 10th, at a Verizon corporate store in Cedar Hill, TX; she lives in nearby DeSoto.
And, unfortunately, from day one, both her Galaxy Note 3 and her husband's presented with the same symptoms that you mention above: distorted over-network voice calls with cut-outs/drop-outs, chirps and popping, sounding more like garbled calls on a bad VOIP set-up. The problems were so prevalent on both handsets that she returned to the original store where they were purchased and where it was verified by the Store Manager that there was definitely an issue on both handsets; he authorized an exchange for both and waived the "restocking fee" since, there were definitely serious issues with the basic "Phone" function of both devices. Hold on to your seats because it gets worse: the two brand new store exchanges she received on October 12th have the exact same issues as her original two handsets that were received for exchange. She called Customer Service last night and conferenced me in on the call since, she suspected that she wouldn't be able to convey her issues since her call quality is so consistently bad. The representative whom I spoke with couldn't have been more professional, after explaining the situation to him and making him aware of the fact that the issue had already been reported by other Verizon customers and was popping up on the web (Google search : "Verizon Note 3 distortion issues or reception issues"), I explained to him that it was the exact same issues that were plaguing the Galaxy Note 3 on the other major U.S. carrier that utilizes CDMA technology and that the issues have not been reported on any of the GSM iterations. After a brief conversation the rep' suggested without hesitation to return to the store and exchange both handsets again. I, of course, was adamant in defending Verizon's network because, I've been a customer since the turn of the good old century and I know for a fact that its' network does NOT have these types of reception issues while engaging i
First and foremost, if you previously had Verizon then, you know it's NOT Verizon's Network.
Just know you are not alone.
Unfortunately, if this is a hardware issue or radio conflict (Google: Galaxy Note 3 new Radio Power Management System), it may take a while to remedy and that's only after Samsung acknowledges that there is a problem. If it's software related, then, that could be a simpler OTA update. Or, it could be a combination of both which makes for a really tragic beginning to this, otherwise, exceptional device. I, too, was going to buy one and was waiting for word on the 64GB iteration but, those purchasing plans are now on hold indefinitely.
Remember everyone: it's NOT Verizon's Network so, decide accordingly. Hope this helps. By the way, the representative whom I spoke to last night went as far as to echo my sentiments and said that, if it's not just an isolated incident with a bad batch then, it was leaning more toward a manufacturer defect.
Typed on my awesome (no voice call problems here) Verizon Wireless Galaxy Note 2!!! 
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