UPDATE!!   I found the solution!  I forgot to enable the Quad/Tri mode in the Subdivide options.  Wow, that was silly of me.  If you subdivide without that option ticked, it won't work, but if you tick the Quad/Tri option box in the Subdivide menu while the tool is active, it will make the same kind of edge cut connection.  The automatic quad creation isn't quite as perfect as the way it's done with 3dsmax, but it works, and works very well!

Three cheers for BLENDER! What an amazing software.