Kashrut (kosher law) seems to say that all fruits and vegetables are kosher, but need to be inspected for bugs and worms (sounds sensible. Eating a bug or worm is not kosher.) There are some specific provisions for grape products, which is related to wine and grape juice, but whole grapes are totally OK.
Leviticus 11, which is part of the basis for Kashrut, seems to only have this to say about plants "And if aught of their carcass fall upon any sowing seed which is to be sown, it is clean. But if water to be put upon the seed, and aught of their carcass fall thereon, it is unclean to you." The carcasses in question seem to be "swarming things" like mice, weasels, and assorted lizards. OK, if I read this right: dead mice can touch the seeds, but better not water the seeds with dead-mouse-infused water. Good. Sounds a little convoluted, but basically sensible.
So what else might make fruit non-kosher aside from bugs or worms or dead-carcass-flavored water? In Israel there are specific laws about harvesting food. Fruit from young trees (3 years or younger) is not to be eaten. I would also assume that the rule about letting fields rest on the 7th year would apply.
So how does one go about getting your fruit or vegetable certified kosher? I wrote to one of the rabbinical councils to ask (they have a hotline!) and hope to hear back soon.
On a side note, I am totally impressed: most of the rabbinical councils have lovely websites.