Various berries have appeared in the anime (even the one on Mirage Island), and we all know that some berries can heal or boost a Pokemon. But I

Certain Pokemon (like Kangaskhan and Tangela) could keep the Berry in their pocket/vines. A couple would have some way of transporting it without hands.

I think that mainly they just don't carry them around, that's what the Trainer's are for

They swallow it, and regurgitate it when needed.

Just kidding. I think their trainers just hold it for them; it's impossible for some Pokemon to ever hold anything. Ditto, for instance. It can't possibly hide the Berry in its goo, can it? Or perhaps Pokemon like Slugma. I think the Berry would just be cooked.

Mmm...roast Berry. Yummo!

Hey, do you think it could get some batter together? Pinap and Nanab Fritters! Yum yum

They don't use Slugma silly!
They use Torkoal for barbecues
and they use Glalie for Soft-Served yogurt (with berries of course!)
aaaaand they use Snorlaxes to squash the berries for juice!
So many berry foods! this could be called the "Berry Recipie Thread"

Anyway, like sNOwBaLL said, they swallow it, reguritate when needed, then swallow it again >_<

Oh.....
Do you want a berry recipe well I have a berry good berry recipe!!
First get a nanab berry,then squash it with a hariyama,
then fish a magikarp and then cook them and then boooooooooooooooooooooooooooooooooom!!!!!
sorry but yeah this is kinda weird