I would remove (you shouldn't need to go in the attic to do so) and take the opportunity to smooth the ceiling

Definitely remove them.

Can you take a picture that has the boxes looking in toward the kitchen?

Definitely remove them. :)

NFI Enterprises crystalized my thoughts exactly. Another Idea might be to make them sky lights.

I would remove them. I also like NFI's suggestion about leaving the moldings and adding pendants.

It is actually just sheetrock over a framework...a means to hang cabinets from a vaulted ceiling. However, it really divided the space and one had to duck down to see anyone on the other side. We have determined after all the votes to remove them. My husband is a paint contractor, and also the builder of the house. It will take some doing, but will be worth it in the long run. We have to paint the ceiling in the adjoining great room anyway, so, this is the time to do it all. Thank you all for all your suggestions!....I am most impressed!!!

Removing them is certainly a viable option, and not a bad one. It is tough to say without seeing the rest of the space more adequately. If you're simply trying to avoid work, then removing them is probably the easiest.

However, you could also try cutting into them and make some shelves. Add a few puck lights, a glass shelf and you would have a cool little display. But thats still more work than just cutting them down.
good luck!

They could be removed and the ceiling repaired in one or two days--I'm guessing. This is the sort of thing that a carpenter or handyman could do fairly easily. There will be dust and that might be the worst part of it. They really need to be gone so why wait?

Whoa, your husband is a painter & the builder of the house!!!! I think you do have the best answer right there, huh? Although I would have liked seeing more photos of the rest of the area.