It finally dawned on me that the memory controller, or a significant part of it at least, is on the CPU rather than the motherboard. And that the Sempron is bottom of the line and the Phenom II X6 is top of the line. I went looking for information about that and found a bit saying that the Sempron and the X2 on which it is based were only rated for 1066, though the X2 could handle 1333.
When I'd first installed the new RAM, the motherboard's autodetection set things to 1333 rather than 1600. The 1333 setting worked and the 1600 setting I tried by manual override was... all crashy. The crashes showed up fairly quickly, too. I could only run Phoenix a few minutes before things went screwy.
Tonight I set things back to automatic and the motherboard decided that the RAM was indeed 1600. I've been running things for a few hours now at that setting and things seem stable. While supposedly the 1333 setting and the 1600 setting are about equivalent due to timing and cycle quirks, there's something psychologically satisfying about running things at the advertised speed.