Microstrip over power plane
Yes, it certainly is possible.  What you do is have some decoupling caps between the two planes, near any place that the LVDS signals cross from one side to another.  The idea here is to give a place for the AC signal return path to cross over from one plane to another.  

Of course, being LVDS, most of the AC signal return current is on the other side of the differential pair-- but not all of it.  LVDS signals are only "mostly balanced".  The less balanced they are, the more return current that will flow on the power/gnd plane.