Actually it's not crucial your pieces be straight with the grain. In clothing construction pieces are cut with the grain or on the bias so they hang right and don't twist. A quilt doesn't have to hang. Plus the quilting will keep a quilt stable.
The main thing you want to watch is to make sure the edges of your pieces don't stretch when you sew them together.
Straight grain is the length. Cross grain is across the width of the fabric and bias grain is the diagonal.
If you look at really old scrap quilts that were cut with templates and scissors the grain tends to be all wonky since they just cut it however it would fit.