What is an efficient way of removing thick grout haze?
You need to keep it wet.  I use vinegar water solution and try to let it sit on the tile for a good 10-20 mins.  Scoop it up and repeat.  If you are polishing it dry you will make it worse (or no better).  

Also next time take a heavy dawn/water mix and rub it on your tile before you grout.  When it is dry then you grout.  And your grout slides off the tile.