That's a hard question to answer without looking at the actual cabinets. The type of wood? The quality of the grain? If it's oak with good grain and you can do a good refinishing job, that might look good. You could then paint the boxes a dark, solid color for accent. Maybe cranberry, forest green, or deep blue. You can also buy hardwood veneer with paper backing to put on with contact cement. (One caveat with that, though: It can sometimes buckle in heat.)
Another thing you didn't mention: Are the doors flush or panel? If flush they're probably wood veneer plywood. It would make the most sense, in that case, to just replace them.
To be worth refinishing assumes solid wood and that you do a very good refinishing job. Even with flat surfaces that's hard to do. It also assumes good grain, which is rare in mass market cabinets. When maple was the fad, Home Depot used to sell do-it-yourself cabinets. I'd have to pick through the doors and drawer fronts because some would have green or even burnt wood in them. And all were made up of poorly matched scraps. They get away with that because most people just aren't very discerning about grain quality. But it still shows in the final product, even if people can't put their finger on why they think it doesn't look so hot. (On the other hand, the self-assemble HD cabinets were very cheap compared to pre-made, and the same quality in general, so they were great for budget jobs.) With newer styles like the dark "mahogany-esque" finish that's currently popular, the wood underneath is obscured by the stain, so it's unlikely to be any good. Probably something like low grade beech or maple. If you can't see the grain well, assume it's no good.
Another issue is type of grain. Oak is open grain, which will show up through paint. Maple will paint better, but the wood is less likely to be worth saving. There are pros and cons with each wood.
Then there's the finish. In most cases that will be some kind of plastic factory finish. It may or may not strip easily. If you don't also strip any face frame then the paint may not stick very well.
Another possibility is that the wood grain might actually, itself, be plastic. I was on a job recently in a very "exclusive" condo complex in Boston. The customer wanted me to remove mirrors that the former owner had stuck to the kitchen cabinets, which were allegedly something like rosewood. (I don't remember the actual wood type.) The cabinets were from Poggenpohl, a company that's managed to market itself as high end. Their store is on the fashion street in Boston -- Newbury St. But the cabinet doors were low-density particle board with plastic veneer. Very tacky. The customer just assumed they were high-grade wood because the condo cost a fortune and the cabinets were bought from a fancy looking "concept" store. Last I saw, the mirrors were still there. There's no way to fix junk cabinets like that. Like Ikea furniture or Scandinavian Design before them, there's really nothing of value in the product, so there's nothing to repair or refinish later. They're just junk particle board with plastic or junk-wood veneers.
If you're uncertain about the doors and they have European hinges, you can remove one of those to see the core.
Without seeing your cabinets and without knowing what level of quality you're going for, I'd suggest sticking with paint for the whole thing if you want a budget job. If you want it nice, and the boxes are "European" style with no face-frame, I'd suggest ordering new doors and draw-face slabs from someplace that will make them to order. Then you'll have basically new cabinets with good wood grain.