There are many ways to cut strips, just find the one that works best with the tools you have. I generally don't use the mat as my guide, although I have checked the grid and it is accurate. I was taught that the reason we shouldn't use the mat in this way is because by cutting on the lines, we would tend to cut in the same places all the time, and this leads to premature wear on the mat, causing grooves or slivering of the mat surface.
I have the June Tailor shape cut ruler and like it a lot. Just be absolutely sure you are in the right slot before making your cut. Learned that the hard way.
Here is a video from Marci Baker, the first several minutes of which shows how she cuts strips. Her mat has no lines on one side and she prefers to use that for less "visual noise". Her method of establishing a straight edge is one I had not seen before.
https://www.youtube.com/watch?v=aQ6oBkgGyvo