This episode of Sick Nick's Strapless Kitesurf Series explains the basics of jumping without straps.
The first type of jump to learn is the basic ollie. Keep your kite around 45 degrees throughout the maneuver, and make your foot stance a little wider than usual.
1) Look for a nice piece of chop to use as a launch ramp
2) As you hit the chop with speed, push down hard with your back foot and lift up with your front foot (similar to an ollie on a skateboard).
3) Now that you're airborne turn the board so that the bottom of the board is pointing into the wind. Also try to make the board as vertical as possible (nose up). This will make the wind hold the board to your feet
4) As you start to lose altitude, point your board downwind and spot the landing. Don't fret if the board starts falling from your feet--you can still pull it off if you concentrate on landing back on your board once it hits the water.
During your first attempts it will be a struggle just to get the fins out of the water.
Once you are proficient at the basic ollie, you can change things up by throwing in a grab midair or sending the kite slightly for more air. You need a lot of wind to be able to send the kite back so that the board stays glued to your feet.
For grabs, first try grabbing with your front hand, then once you get those down try some with your backhand.
Remember to do everything the same during a grabbed jump as a regular jump...you still need to position the board into the wind to keep it nearby (hopefully on your feet).