The thing is that you drop things where it is most convenient and that's on top of the desk. If you had a desk hidden in the cabinets would things get dropped on top of the counter or would you actually open the cupboard to drop things.

If you had it as part of the cabinets it would be too high Because counters are higher than desks. If it weren't then you'd have to pull it out somehow and that would be very inconvenient.

I think your best bet would be to get some attractive baskets to drop things into. Perhaps an in-- out office tray would work for your mail and a small matching basket for other stuff-- or a couple of drawers set aside for these purposes. I do however suggest that you keep the mail separate from the rest so you can deal with it without searching.

Ours is not hidden, however, my dh made a cubby that fits directly under and attached to the cabinets. It's a mail slot made from oak, so all the paperwork goes in one of 5 slots and the unit matches our cabinets. Each slot is about 8.5" wide and about 3" high, and is as deep as the cabinet (about 12"). It keeps the desk clean and paperwork filed until it can be dealt with.

It beats having it all enclosed, because just like the previous poster said, it wouldn't go where it was supposed to because of the hassle of the doors.