Well, I think I might have fixed it on my own. Can't be 100% certain, but I can say what I did.
The last update to my AMD drivers installed this Gaming Evolved program, that had stuff like game optimizations and a Raptr plug in. I disabled that and set it to not start with windows. I also found the ME3 configuration utility and made some adjustments there as well. I have not seen the stuttering problem since. So it may have been one or both, I can't be sure, but I hope that helps anyone else.
I also just disabled that Origin overlay as well. I don't really need that, so that was a good suggestion, thanks.
Just for reference, these are the ME3 configuration utility settings I am currently using:
Windowed: Fullscreen
Window Mode: Normal
Aspect Ration: Widescreen (16:9)
Resolution: 1920x1080
Anti-Aliasing: On
Motion Blur: On
Dynamic Shadows: On
Number of Cinematic Lights: Default
Use Spherical Harmonic Lighting: On
Anistropic Filtering: 16x