I do not find it necessary at all to tear fabric for quilting. I do not prewash fabric so there is some starch/sizing in it that keeps it crisp, and I agree with the poster that the *fold* is what is important. Having all your cuts on the correct straight-of-grain is very important in garment sewing because it affects the hang of the garment. It is not that important in quilting as long as you are not stretching the edges of your pieces out of shape. That is where having some starch/sizing in your fabric helps. (If you prewash, then it's a good idea to starch the fabric before cutting.) For quilting, all you need is to be somewhat close to on-grain; it does not have to be perfectly on-grain.
Be aware that when you tear, the fabric is damaged for up to 2 inches from the tear. This damage may be invisible to the eye, but becomes apparent under a microscope. You really do not want seams within that area of damage. Also, tearing cross-grain is more damaging than tearing on the lengthwise grain. Tearing on the cross-grain often distorts the fabric edges. I will sometimes tear long borders on the lengthwise grain (parallel to the selvedge) because it is easier to be accurate on long pieces; however, even then I trim off the edge that was torn to avoid using damaged fabric.
Edit: I just want to add that being completely square is not the same as being completely on-grain. Being completely square is a geometric measurement. You can be completely on-grain and not square if the grain is distorted (which is often the case with fabric off the bolt). Likewise you can have completely square cuts without being on-grain at all (as in the case with bias cuts).