Okay, so I rolled back to Maya 2010 for this scene and the grraininess went away. I had a ton of issues because all the saved versions I had had been resaved in later versions so I had some issues.. Even after I fixed those, renders would occasionally lock up and freeze, and batch renders would only do one frame. I noticed that it only happens when a certain area within the scene is visible in the render.
I did a bunch of troubleshooting by deleting one object at a time, and by rendering different views, trying to find the troublesome objects, but it seemed to be more of a cumulative thing. It's progressing nicely in a batch render now. At first, I could only get through a few frames, so...I SUSPECT, the problem has something to do with the fact that my CPU (Ryzen 3600) seems to be overclocked a little (4Ghz), but the V-core looks to be stock. Think I'll reset factory defaults and try again. Or maybe bump up the voltage a little. I guess at some point, I may have half-aXXXd a stress test.
First, I'll see if this render completes. If not, I'll remove the overclock and restore the deleted geometry and textures.