Cut a in skylight or two:

Here is a link that might be useful: Velux

I would add above cabinet lighting, too.

Natural=Skylights
Artificial=Dimmer controlled lighting on the tops of the
cabinets.

Is it possible to install normal ceiling 4-inch j boxes? Just what you'd install to hang a pendant light or chandelier or any other ceiling light?

This:

We also could not install can lights, and our city inspector suggested we look at recessed LED lights that fit inside ordinary j boxes. It took some research, but we ended up getting these:

http://www.homedepot.com/p/Unbranded-6-in-Soft-White-LED-Disk-Light-For-Recessed-Can-Lighting-CE-JB6-650L-27K-E26/203886372#specifications

It looks basically exactly like a can light when installed:

Here is one partially installed without the light diffuser installed over the light:

After you connect the wires, you just screw the whole white thing that is hanging down directly to the j-box (the screws pull it flat to the ceiling, and then snap on the light diffuser.

It looks great. My only complaint is the color of the light they give off is a little more yellow than I would have liked, but they are bright, and they dim, and they don't buzz, and they didn't require the giant contraptions that can lights usually do, and we're happy.

Here is a link that might be useful: The lights we bought

I'll look into the j-boxes, but I am pretty sure we cannot even do that. Currently we have one chandelier hanging in the kitchen (you can sort of see it in the picture), and we will be using that box to base the track from which to hang pendants over our island.

I think the problem is that there is no access to place boxes anywhere else in the ceiling. You would have to cut out the planking to add the boxes, and how would you run the wiring to get to the boxes?

I love the idea of putting in skylights, but it just isn't in the budget right now - and I am worried about leaking as well. This whole project was triggered by ice damming from last winter - the ice melted into the kitchen and down the exterior wall into the basement.

We will look into the above cabinet lighting. Any suggestions on type? More LED?

What about track lighting systems? perhaps one on the wall side of the ceiling next to each beam?

In the picture, you can see we have a track on the bottom of the middle beam already. The track heads are not the best (plenty of light, but blinding when sitting in the living room). This picture was taken on a very sunny day - few and far between here.

"We will look into the above cabinet lighting. Any suggestions on type? More LED?"

That my preference, I promote the JUNO T12.

I'm not a dealer. They are leaders in R&D.

Another alternative, would be to coffer the kitchen ceiling, which will allow for the placement of fixtures and elec. conductors.

These may work well in your application:

Here is a link that might be useful: Remodel cans

Trebruchet and Jillius - we cannot use cans at all. There is no space to put even 4"cans. Everything has to be surface mounted.

We are putting LED tape under our cabinets, and LED hybrid rope above the cabinets (only on the wall shown in the first picture above).

I think we still need more lighting for the room in general. Would running more LED rope next to the lower beam provide enough lighting?

" Would running more LED rope next to the lower beam provide enough lighting?"

Highly unlikely. Skylights.