I noticed a distinct "thunk thunk" coming from my bottom bracket area (bottom knocking? ), looked down when it was happening, and noticed the chainrings moving in and out maybe 1-2 mm (at most) in time with my pedal strokes.

Got home, turned the bike over, and the crankset does move a bit when torqued side-to-side. The bottom bracket cups are solid in the frame - they're not moving at all.

The bottom bracket and crankset are Ultegra 6700, probably 10 or 12 thousand miles on the bottom bracket, less than half that on the crankset. (I could figure it more closely, but that's probably close enough)

Time for a new bottom bracket?

Is the spindle sliding along its axis? If so check the LH crank is properly attached to the spindle: unscrew the small pinch-bolts, tighten (gently) the wide fixing bolt that pushes the arm onto the spindle, and re-tighten the small bolts.
The wide fixing bolt needs a special tool, "TL-FC1":

If that doesn't help, remove the crank completely: remove the 3 bolts mentioned above, pull off the LH crank arm, and slide out the spindle with the RH crank. Check you have all the appropriate spacers by comparing with the diagrams from Shimano Techdocs: Double, Triple, Compact

Once you have removed the cranks the state of the bottom bracket will be clear. If the cranks are ok but you're getting 1-2mm of movement, either the threads or the bearings of the BB must be utterly shot.

Before reassembling the cranks, ensure that the splines and spindle are impeccably clean and given a thin coat of fresh grease.

Spindle isn't sliding, and the LH crank is tight.

Guess I'll be yanking the crank.