The USDA/NRCS web sight says persimmon can be propagated from cuttings, but not all cuttings are created equal. Persimon can be started from cuttings, but it's difficult. I started pears from cuttings, and got two pears from about a dozen cuttings. So with Persimmon plant lots of cuttings, expecting most to die.
Two things I noticed from your pics. #1 there should be 2/3 of the cutting in the medium, so the top doesn't dry out. #2 the hood, or a clear plastic bag should be used to create a humid/greenhouse effect.
Persimmon has no anaerobic tolerance, which means it doesn't survive in saturated soil or flooding. That tells me that the cuttings should be in moist, but not saturated soil. Rooting cuttings is a balancing act between rooting and rotting, and sand is less likely to cause decay than soil.
Some cuttings will leaf, but that doesn't mean they've rooted. When plants leaf, you're halfway there, and humidity is important when leaves appear. Without humidity or a mist system, leafed out cuttings are likely to die, because the leaves are drawing from a cutting that's not rooted.
When a plant can be rooted from cuttings, that doesn't necessarily mean hardwood cuttings like we discuss on the forum. Cuttings can be hardwood, semi-hardwood or softwood. It appears the best success with persimmon is semi-hardwood. Here is info:
http://en.allexperts.com/q/Seeding-Propagation-733/Wild-persimmon.htm
Good luck, and I hope to see success pics. :)