One of the questions friends and family ask me a lot is how to clean a grinder, usually because it's something I might have mentioned doing, that hadn't occurred to them. The truth is, though, most grinders aren't very good at keeping themselves clean, and part of your owership routine should be regular maintenance, including cleaning. The best part about it is, you've probably alreay got everything you need to keep your grinder's innards looking fresh!
Minute rice, or similar*
A brush, or a cloth
A vaccuum cleaner with a hose
The first step I usually take is to just vaccuum out the hopper, the grinds bin, and anything that looks like it's collected some debris. Nothing in-depth, just a quick pass-over with the hose to tidy things up a bit. Then, I start my grinder up, and add about 1/4 to 1/2 cup of rice. I usually start with a fine grind setting, letting about 60% of the rice pass through, and then I move coarser, to more of a coarse drip setting. I find that the fine grind really helps clean up the dust and oil, while the coarser grind helps push everything out. You can discard the ground rice, or maybe like, bake it? I'm sure there's some creative use for it, but I just toss it out.
Moving on, what you'll want to do is remove the hopper, and take out the top burr carrier (though, if any of this voids your warranty, don't bother!). Using the toothpicks, scrape out any compacted coffee grounds wherever you find then, and vaccuum up the mess. You can use a nylon-bristled brush or terrycloth to clean out the grinds chute, and break up anything a toothpick has trouble with. Once you've vaccuumed all the residual stuff out, you can replace the burr carrier, wipe out the hopper and bin with a damp cloth or paper towel, dry, and re-assemble. I will typically run an ounce or so of beans through the grinder again, to "re-season" it, basically preventing rice from showing up in my next brew.
I do this probably once a month with my Vario, or more often if I'm expecting a drastic change between beans. Recently, I've let my sister use my grinder for her french press, and I've noticed that her more oily, dark beans are tainting my lighter, more delicate beans. So, I've been cleaning a bit more often, about as much as I notice a distinct rubbery aroma in the hopper.
As I stated above, don't go too far mucking about with grinder internals if your manufacturer doesn't think you should. If you're not supposed to remove your burr carrier yourself, you can just run the rice through, vaccuum what you can, and run some beans through.
* Minute rice is often the more accepted type to use as a grinder cleaner, but I have used a variety of other rices with no visible damage. I won't go so far as to say "jasmine rice won't hurt your grinder", though, as my experience is only limited to my Vario - which caught a stone not too long ago with no visible damage. It's a robust grinder, I suppose. I would say that with most grinders, it's best to start with Minute rice, or a similar quick-cook brand, before venturing into anything denser. There is, of course, the Grindz option, and I know a few folks who swear by the product. If the idea of ground rice makes you nervous, Grindz is a perfectly reasonable, purpose-made option.