Way back in the day, when I played a little LB/SS, we were taught to slide to our job side (right or left depending upon responsibility) on a cut block, using our hands to push the defender down and away from the job side, while keeping our eyes up because a cut block likely meant the play was heading our way. Of course, keeping your eyes up makes it more likely to miss with your hands.
I received a pretty bad knee sprain from a cut block and quickly figured out that what we were taught was crap. We were taught that a proper cut block is going for the knees but not trying to injure the other player. I decided that a cut block was trying to take out my knees and you didn't care if you injured the other player. So, I came up with my own technique.
Honestly, it probably was technically illegal but I never got flagged and only recall one guy trying to cut block me multiple times in a game (though, it's a small sample size). I figured that if you were going to try to hurt me I'd try to hurt you. Instead of sliding to the job side I would sort of hop and try to land my cleat in the guy's ear-hole. About half the time his helmet would come off. I'm guessing, from a distance, it just looked like the cut block worked. But the offensive player, usually a FB/HB, would often try to get in my face after the play about it. I just said, "Go for my knees again and it'll be worse" - and left it at that. They usually went in higher the next time they blocked to my side.
The one guy I remember that tried to cut block me multiple times I caught really good the third time, I think, (maybe the fourth) and he came up swinging after the play and got ejected.
Now, I'm sure my solution would not work today, with replay and the extra emphasis on safety and all, but that was the only injury I had playing defense.