First of all, I line up the selvages and press a new fold, then I cut my selvage off. If I am cutting lots of strips, I double up the fabric end to end so I have four layers.
I have drawn a line on my cutting mat about 1/2in from one side. I lay the fold on that line, that way I can see the markings at the sides that I can line up my ruler. I cut from the folded side.