I am right handed. If I am cutting something that can be a little off (HST's that will get squared, strips that are templates-think Judy Niemeyer), I may rush and cut with the bulk of my fabric to the left, using the guidelines on the cutting board. If it has to be "exact" - I turn the fabric so that the bulk is on my right and my ruler stabilizes the piece I am cutting.
I was cutting in the "exact" way in a quilt shop one day while I was taking a class. Another quilter came up and asked me why I did it. I told her it made me more accurate and gave me a chance to correct a mistake if I mis-cut rather than have to scrap a lot of fabric. First words out of her mouth....."Why hasn't someone told me this before?" Ironically, I figured it out myself because I was so good at making bad cuts that had to be squared up.