I used minky backing for the first time on my last two quilts. I did a simple 3" grid like you are describing.
The thread that I used made alot of difference for some reason. The first quilt I used Superior Bottom Line (top and bobbin), and I had frequent birds nests on the back. I tried every tension trick that I could think of to no avail. It was very frustrating.
The second quilt I used Superior SoFine without any problems whatsoever. I didn't have to change a thing to get perfect stitches.
I did notice portions of the dark minky backing being pulled up to the front on light fabric... especially when I used a larger needle on the first quilt. I went back to a smaller needle on the second without a problem.