Thanks guys for all the advise. After reading your posts and consulting with a friend who has first hand experience renovating buildings, it was suggested that I leave the old layers of tile down. One reason is that the original 8" linoleum tile (circa 1956/57) may contain asbestos, so leave it alone. Another was the irregular surface left from pulling up 2 layers of tile/linoleum with the old underlayment of tar paper being ripped up in spots.
We put down a 1/4" underlayment of Masonite/masonboard? [this is not the same heavy duty stuff I remember from years ago when Dad remodled his home.], nailed it with wood nails (those funny nails with twisted edges), and proceeded to lay the linoleum tiles on this new FLAT spiffy surface. This new surface really made things easy to install. The only problem area would be if you had to go under a door, as the extra 1/4" thickness could cause a problem. The only area I couldn't get to is under the dishwasher, so when we pull out the sink and dismantle the dishwasher, we'll have to figure out if we want to refloor that area, as it's "raw", showing the original floor boards of REAL solid wood [2"x8"?].
FYI: We're doing half the kitchen at a time, as we have to still LIVE here. The cabinets are being made by my husband and we'll shift from one side of the room to the other, hoping the sink, et al, will go in speedily.