Every coach and kettlebell lifter will likely tell you something different about how you should grip the kettlebell, whether it's thumbs forward or thumbs back.  As with any other part of your technique, this depends on your body type and personal preference, so I can't tell you what is "right" and what is "wrong".

I can, however, tell you my personal preference for each lift and my reasoning behind it.

For One Arm Long Cycle (OALC), I prefer thumbs forward. Since I don't need a long backswing in my clean, especially if I'm going at a fast pace (10+ repetitions per minute), keeping my thumbs forward allows me to clean the bell faster. Additionally my elbow stays in tight to my body allowing for the pop from the hips to launch the bell into the rack position.

For Two Arm Long Cycle (TALC), I also use thumbs forward. When I go thumbs back, I find myself rounding my lumbar spine and knocking the kettlebells together, making my clean less efficient. With my thumbs forward, I find it easier to keep the bells on a straight path and use my hamstrings in the backswing. If you look at the top male Long Cyclers, you will see both thumbs forward and thumbs back - so choose what feels most natural to you.

For Snatch, I go thumbs back. I find that I can get a deeper backswing when I let my hand turn, which becomes crucial with the heavier bells. A long backswing is essential to maintain the pendulum swing, which will make the Snatch more efficient and less taxing on your grip.

What grip do you prefer: thumbs forward or thumbs back?