The trick of sewing off the little piece of fabric works well at the beginning of a seam. But what works,too, is getting a 'single needle plate' for your machine. It has just a small hole for the needle to go through, not the oval shape that comes with the machine. It keeps excess fabric from getting sucked down underneath. You have to remember to only use it for straight stitching, though, or you'll break a needle.
If the fabric is being pulled under, you might not have the fabric far enough under the presser foot so that the feed dogs catch it.