After I do the 'lining up' thing with the selvage and before I fold in half again (fold to selvage) I hold my fabric by the pressed fold and shake a little bit. This helps the fabric get the true grain. Then I lay on the cutting mat, fold in half again. I also am able to cut with both hands and never use the lines on my cutting mat. I cut my first strip with my left hand with the ruler line on the fold and the edge to cut, then switch to cutting with my right hand and do the same thing. I line up one of the lines on the ruler with the fold line and the cut edge with the width I need to cut my strip/block.
Clear as mud? I'll try and get some pics up this weekend to help.