Hi Chandler -- Thanks for that, and I did get them off. The trim rings were so snug that the screws did pull partly out of the wood (well, OK, the very thin paneling that will be wood when it grows up).
But mission accomplished, and nicer looking speakers installed. Now that I get how Damon uses air space under the cabinets, I'm going to fish a wire from the over-stove fan through the kitchen cabinet false bottom for an over the sink light.
It's no fun if nothing needs tweaking. Thanks again.