They have a very very mild flavor, reminiscent of pear or apple, but not as strong. They also have a tendency to be dry (as in not very juicy at all) it depends on the quality of the fruit of course. Something about the texture/mild flavor reminded me of squash..
A few weeks ago, I was watching some donut game show on Cooking Channel. One contestant chose to use persimmon, without being aware how it tastes. The person mentioned that it was similar to mango.