Have you looked at the under cabinet LED or halogen puck lights? You can have these recessed into the bottom of the cabinet.

Well, the bottom of the cabinet is 5/8 inch thick plywood. Is there really something that can be recessed into that thickness of bottom shelf? I would consider it although I suspect I would still be able to see the glare of the lights when sitting at the table. Any suggestions for a product that would fit the bill? I am also interested in anyone's opinion who believes I can get away without under cabinet lighting given my upper cabinets only project 11 1/2 inches over the counter (I think this is less deep than ususal).

They sell LED lights that are much thinner than tubes - even the thin tubs. They are very expensive though.

If you have no lip to hide them behind, anything you put on the bottom of the cabinet will be visible. If you want undermount lighting on old cabinets, that is the trade-off.

If the cabinets are painted, is there any way to put a small piece of molding across the bottom to "hide" lighting behind? Obviously, it is a lot easier to match paint than stain.

I have thought about adding a lip to the bottom edge of the cabinet, however the uppers are already so low I can't afford to lose more clearance (there are wine bottles that I can't push to the back of the counter--too tall). I think for now I will proceed with ceiling lighting and see if I can't obviate the need for UC lighting by careful placement and choice of fixtures/lamps.

Try looking at Kichler Lighting. I think they have very slim profile lights. Good luck.

costco is carrying LED battery operated puck lighting. They are about 1" with a 3" diameter and have adhesive backing. They come in white, and I think metal. 3 for 20.