Use a June Tailor Shape Cut Plus ruler. Here is a link to one on Amazon:
http://www.amazon.com/June-Tailor-12...dp/B0001DUMTA/
All you need to do is line up the fold of the fabric with a line on the ruler. The smaller Shape Cut is not as good because you need to fold the fabric twice and line up both folds. With this ruler, you can stack fabrics as long as the folds are lined up an equal distance from each other and at least one of the folds lines up with a line on the ruler.
The slots really help with keeping the cuts straight. What determines whether or not you get the dreaded "V" in a strip is how accurate you are cutting 90 degrees from the fold. It actually has nothing to do with pulling threads or cutting on the straight grain of a fabric. You can cut straight bias strips from fabric -- which is about as far from on-grain as you can get!
Edit: I should add that cutting exactly on the grain is not actually that important for quilting. The only reason to be concerned about grain is if your strips are going to distort while you are sewing and ironing. If I heavily starch fabric before cutting, I find that even totally bias cut strips do not distort on me while I work with them. Grainlines are much more important when sewing clothing, as it affects the drape of the clothing.