Dawn and a soft cloth. No Magic Erasers or anything else abrasive.

And fix the ventilation issue that led to the problem.

that seems odd.....I wonder if the previous occupant actually did something to the finish thinking they were evening out imperfections or worn areas, and it wasn't the right thing/didn't cure right because of the substrate and you have a gooey,residual-tac type mess which may not really be grease. Grease on ALL surfaces sounds peculiar-i'd analyze this a little more. You could strip and restain if need be, but I'd want to know really what I'm dealing with. Greasey fingerprint areas around the arm reach height is what you'd expect, but that's about it.There is something called "Kotten Klenser"-a pastey gel that can be smoothed on/gently left various amts of minutes depending on extent of buildup-I've used it and like it. Can be done initially for a min or two with option of 2nd application for more removal-it will remove even down into the finish if desired. this might be a good product if you don't know what you are dealing with....it is flexible and adjusts to your application and timing. I found it at an antique shop-the lady uses it on all pieces for gentle cleaning and even for more aggressive cleaning , if need. Any product/procedure you choose should be done on [out of sight] test area....I just get a red flag feeling about "grease all over"....

Plain mineral oil cuts through kitchen grease instantly. Of course, you'll have to use some good microfiber cloths to wipe off the mineral oil, but that isn't hard.

Very gentle, no chemicals, no abrasion or scrubbing.

"Plain mineral oil cuts through kitchen grease instantly."

shanghaimom, do you mean mineral spirits or mineral oil?

Mineral OIL, for sure. I have some that I use for cutting boards, but it is the same as the unscented stuff you can buy at any Walgreen's in the medical supplies area. Like Baby Oil, without the scent.

I'd be nervous about trying mineral spirits--wouldn't that strip any finish off?

I don't think so. One of my cabinets was delivered with something sticky on one spot, maybe the remnants of a label or something, and I used mineral spirits to remove it. I did work quickly, though, and removed all traces of the mineral spirits with a well wrung-out rag of warm soapy water. No effect on the finish, but I don't think I'd want to try it over a large surface if something milder would work.

Interesting about the mineral oil. I will have to give that a try in the future. I would much rather use that than a solvent.

A "layer of grease" takes serious inattention.
With mfg cabinets like Kraftmaid and most any with a catalyzed varnish very few materials affect them especially in small amounts. Mineral spirits will not attach the finish used gently. (even acetone or gasoline are on the list of chemicals that the finish resists) Paints can be more delicate.

One issue that is not well know is hand lotion. I've heard of several cases where clients that use a "lot" of hand lotion have softened the finish around the hardware. I've mentioned this to the missus who wipes hands if she's been using it, not that it is excessive, just being careful.

I have seen "Guardsman" specified in several mfg literature. Worth looking for, or check your brands recommended maintenance, usually on the website.

Against medical advice I regularly use a light duty orange cleaner on mine without issue.

That said manufacturers do NOT recommend most furniture polishes or things like "Liquid Gold", Murphy's oil, or most lemon thingies.

Thanks for all the great ideas! Yes, I am going to address the ventilation issue; the former owners didn't have an exhaust fan, only a microwave with a recirculating fan over the stove. I'm going to replace that with a Kobe range hood. Unfortunately I can only use a 30" hood over the 30" stove because there are cabinets on both sides and the kitchen is small anyway... can't sacrifice any of the existing cabinets for 6" more of range hood width. The former owners were meat eaters and I'm a vegetarian so things should be fine from now forward, LOL

I think the problem is that they lived there for 10 years with no real ventilation and never once cleaned the exterior of their cabinets. :/

I will try the mineral oil first, because I already have some on hand (I use it on my maple cutting boards). I have fragrance and chemical sensitivities and so it's a big plus that it is odorless!

Now if someone only knew how to remove "dog stink" from the garage (see my post here http://ths.gardenweb.com/forums/load/cleaning/msg0708291313765.html which sadly no one has responded to)!

Mineral spirits will NOT strip any finish off. It's my go-to liquid for removing greasy crud and grime from old furniture and cabinets.

You can get "deodorized" mineral spirits. Works just as well.

Wear gloves and ventilate well.

Another vote for the mineral oil here. I have it handy in my kitchen for my butcher block tops and cutting boards, as well as my Soapstone counters. I have found it to be very effective at cleaning grease and fingerprints off almost every surface in the kitchen. I use it to wipe down my grates and burner bowls on my Bluestar range, it also works great on stainless steel and the painted part of my range. Works great on oil rubbed bronze hardware to keep it looking, well, oil rubbed! Love the fact that it is food safe, oderless, and chemical free.

I was curious to hear what KraftMaid would recommend, so I called them (expecting them to try to sell me on some kind of proprietary cleaner, LOL)... and was told that the only thing they recommend for their cabinets is Murphys Oil Soap.

Now curious, I went to Murphys web site to see if the main ingredient would be mineral oil and perhaps some surfactant (soap). The ingredients are shown as: "Specially formulated with: water, coconut & plant-derived cleaning ingredients, natural fragrance, and 2% synthetic ingredients". Their logo says "Pure Vegetable Oil".

Knowing that mineral oil is NOT a vegetable oil but is a petroleum by-product, and also knowing that vegetable oil is not recommended for cutting boards but mineral oil is, I did some research into why oils would work to clean off grease and oil.

I found this interesting explanation at "The Manly Housekeeper"
http://www.themanlyhousekeeper.com/2011/05/18/quick-tip-the-best-way-to-clean-oil-splatters/

and in the process it explains in an update the performance difference between using vegetable oil (in Murphys) and mineral oil:

"Although vegetable oil works fine, a chemist reader of mine explains why mineral oil is the better long-term cleaning choice. âÂÂShort-term, vegetable oil does work just as well, for the shared common property of âÂÂoilinessâ that youâÂÂre taking advantage of here. Longer-term, mineral oil has the benefit, over vegetable oil, that it will not âÂÂyellowâ and will not get tacky or gummy, as will vegetable oil. The reason that vegetable oil is good for your diet �" polyunsaturated fatty acids �" is also the reason that it goes yellow and gets sticky with prolonged exposure to oxygen in the air.â "

So from the information above, it seems that although KraftMaid recommends Murphys Oil Soap, it sounds as if the surfaces might remain cleaner longer if mineral oil is used... UNLESS the other ingredients contained in Murphys happen to counteract the propensity of plain vegetable oil to go yellow and get sticky over time.