When I was taught to sew back in the 70s it was beaten in my head to straighten the fabric and cut on the grain. I still do so, you tug on opposite corners if it is a manufacturer's defect or I do typically tear one edge. I've found that almost all fabric I buy is cut terribly off grain and I'll typically lose at least 2" on the wide end. For a good tear you need to have at least half an inch on the side you start the tear. Back when I was a child I was taught to pull a single thread out and then cut along that line but I can't see that any more!
I don't have really any experience with flannel so can't say anything directly about how it tears.
But as the Gnome said, it's not that big a deal any more. Most fabrics are solidly woven and using a proper machine stitch tension and length (small stitches for quilting) it isn't an issue. I've done enough fussy cutting over the years paying no attention to the grain lines and it all goes together well. Some people are a big believer in starching, I don't bother with that, just try to sew each piece "gently" and remember to let the machine do the work. If we are tugging and moving the fabric we are skewing it and building in ripples -- just let the machine do the intake and gently guide it into position.
One thing I have noticed, I have a lot more loose threads on me and that need to be removed from the quilt back than most people I know. I think that is a direct result of me being so true to grain. I'd like to give myself a bit more freedom but old dogs and all that!
One way to see if it is true enough is to cut a strip and then open it up to look at it. If you can't tell where the folds were you are good enough! If you get deep V's at the fold marks, not so good -- but of you are subcutting and it doesn't matter then it doesn't matter.
As a general tip for using the mats and rulers, you really don't want to use the lines on your mats they will wear out faster where you typically cut. Turn the mat over and use the back to cut on. Use your rulers (and it helps if they are all the same brand). There are videos that show the techniques much better than I can describe them!
Being right handed I have two basic choices, one is to cut going from left to right so my ruler shows my cutting line and is held down with me left hand, or I often cut from right to left using two rulers. One ruler measures the depth of the cut, the other ruler is the standard 6 (or 6.5) by 24" ruler. I slide the right ruler along the width I want, positioning and stabilizing the long ruler with my left hand. The second technique really works well when you have cuts wider than 6". If you have a fabric that is too close to the color of your mat, slip a piece of paper under the cut edge about half an inch. Obviously if you have a white mat you use a colored paper, if it is Olfa green white typing paper is awesome.