I've used 3/32" luan mahogany plywood. Ring nails at frequent
intervals, etc. Fill nail spots with concrete patch type caulk, and
smoothed out with a putty knife. Otherwise, nails will "print
through." To deal with the squeeks, use deck screws before you put
down the plywood. This all does raise the floor a tad, but I doubt you
will have any problems except where you run into a carpeted room.
There are some various wood or metal transition strips you screw or
nail in place. I have also used wedges driven in and glued from
underneath when I had a problem by the front of a refrigerator in a
kitchen where there was a lot of foot traffic and stress on the floor.