I ALWAYS straighten the grain on my fabric before cutting... that is unless the print on the fabric (like a stripe) is way off the grain line and would make the quilt look "caddy-wampus." I think it is very important to keep on-grain for the borders especially. It just make everything lie flatter and quilt easier, either by hand or by machine.
When my mom taught me how to sew, she taught me how to "pull a thread" to get the straight-of-grain line, and then how to pull the corners to square up the fabric.