I've never heard of anyone been successful rooting persimmon cuttings. You can take root cuttings and they will root but not a branch. The common way of propagating it is grafting into desirable rootstock in spring.

Bass

You can learn how to graft. There are also many places that will custom graft for you. It's not expensive.

Also, some states have scion exchanges in the spring, and there will be people sitting there doing custom grafts for the public. Again, not expensive.

Depending on the size of the cutting foliage stage of growth and area the cutting was taken from you can root just about anything with the right tools and care. It's easier to root a clean cutting that is in it's growth stage and at least 5â in length with a small portion of it hardened off. On the easiest specimens just soaking them in clean water with just a few drops of hydrogen peroxide, (Partial shade, rooting portion dark, cool, 50-60 deg for most and water changed daily) will do the trick. The more difficult ones need cloning gel and cloning machine or a type of aeroponics assistance to prevent rotting or drowning in the root zone. If it has too much foliage it will dehydrate fast with out some no-wilt on them. It is also important not to give it too much nitrogen as it will inhibit rooting, instead use something that is formulated for early root growth. My best results come from a aero/hydroponic machine using beneficial fungi and bacteria (no peroxide) like House and Gardens Roots Excelurator.
Hope this helps

I've never heard of a Japanese persimmon cutting being rooted. I doubt that it is impossible, but I would bet that it would not be easy or even do-able with just a glass of clean water and hydrogen peroxide.

I've never tried this myself, and it is true that techniques for rooting persimmon cuttings are not well known and not generally practiced, but I have heard a claim (that sounded legitimate) from someone who said that they were able to root persimmon cuttings. I don't recall the process, but I do know that it was fairly complicated. In any case I'm not sure that there would be a great benefit to having persimmons on their own roots, as there might be for say pawpaws, which are more likely to be grown in single clonal groves and so benefit from having the pedigree root genome.