I have to (more or less) agree with the general consensus here...for the most part, anything like a UV filter will only degrade your images and do little or nothing to actually protect your lens. Point blank; if you drop your camera on the lens hard enough to break the filter, you've probably broken the lens as well....ESPECIALLY with a kit lens. It may not scratch the glass, but chances are you've jammed or broken the gears/mount/focusing mechanism. I have 2 not-so-proverbial paper weights I can show you, a Tamron 70-300mm from my Canon days that fell out of my pocket and a Nikon 18-55mm that took the hit when the camera fell off a chair. The Tamron ended up with a bent mounting ring and damaged e-tronics and the Nikon is now in 4 or 5 pieces. In both cases, a filter would have done NOTHING to protect either of these lenses. Seriously...as far as protecting a camera lens, you'll be FAR better off just learning how to take proper care of your camera and learning how to properly clean your lenses. I won't lie to you...yea...sometimes sh!t happens. My Nikon D90 recently took a tumble while shooting water falls out in Hocking Hills, Ohio...the lens survived, but the camera's diopter and flash are busted all to hell. I was farting with the tripod and the camera came loose...normally I have the camera strap wrapped around my wrist, but this was the -1- time I didn't and fate bit me in the tush for it. That said, again something so trivial as a lens filter would have done NOTHING to prevent this.
Back in the days of film, filters were pretty much a must. For example, if you were using day light balanced film and needed to shoot something indoors under incandescent, you either had to stop and switch rolls of film or you used a filter. As Derrel said, film also had a tendency to be sensitive to UV and such, so again a filter was the easiest cost effective solution. Don't even get me started on b&w film with filters, LOL! Today however, digital cameras have gotten very good at things like color balance and such and using the likes of image editing software (Photoshop, Lightroom, Gimp, Coral, etc) you can easily compensate for anything the camera may have missed. Today UV filters really are little more than just another way for the sales guy to up his profit margin a hair.
As far as "dirty/greasy...airborne food particles", while I don't shoot food specifically, as an artist and musician, I spent A LOT of years on a cook's line...I was "in the service" (as in "food service") on and off for around 20 years. In short, most airborne grease comes from the cooking process. Yes, you can see A LOT of built up grease on kitchen walls, in vents/filters, etc.. That said, the solution here is easy - DON'T SHOOT YOUR IMAGES ON THE COOK'S LINE (or the prep kitchen if said restaurant has one). Most restaurants have some form of vent/exhaust system that literally sucks most of that grease right out of the air, so if you take the food out to the dining room (or some place isolated from the kitchen/line), there's very little chance of having to deal with significant amounts of grease while you're shooting. At worse, you may get some steam if the food is still hot, but that's easy enough to wipe off with a clean soft lens rag (a lens cloth is a great accessory, however an old soft t-shirt works well too). Seriously - you're using a flash setup anyways...just take everything out to a table on the far side of the dining room.
Now with all of that said, I do agree with dennybeall in that depending on what you're shooting, a GOOD polarizer can be the exception. Often freshly prepared food substances will have a degree of moisture that can cause some reflections. Much like shooting water or anything else that produces reflections, a GOOD polarizer can reduce this...HOWEVER...my suggestion would in fact be to try it both ways. You could very well find that those little reflections actually add a degree of appeal to food photography as it adds to the illusion of "moist and juicy" (so to speak). By using a polarizer with certain foods...say a hamburger patty for example, you could potentially make the food look dried out, which most would find unappetizing. In other words, try it, but use some discretion.
Last but not least...and I say this in all earnestness - get used to the idea of cleaning your camera gear! The simple fact of the matter is that unless you're going to keep your gear locked up in a hermetically sealed bubble and NEVER use it, your camera and lenses WILL get dirty occasionally! It's NOT something to be paranoid about. Personally I live in a house where my wife and I have 3 dogs and 2 cats AND my wife and I are both smokers, so even when I keep my gear in a closed case, I -still- have to clean things quite regularly. Whether you shoot food, landscapes, sports, etc., sooner or later you lenses, camera body and even your sensor WILL get dirty. Very simply, cleaning your gear goes with the turf.
Just my own opinions...use them for what they're worth.