Microwaves have a much larger wavelength than light waves - while the light waves fit right through, the microwaves end up hitting the metal mesh.
I wrote a Staff Report on this very topic.
Microwaves big. Light waves small. Same reason butterflies can’t get past your screen door, but gnats and no-see-ums can.
And of course if you were trying to shield yourself from Hulk-inducing gamma rays by hiding behind your appliances, you’re out of luck.
Q.E.D.
Ok, I guess micro waves are a loot larger than they sound from the name.
But that raises another question.
So the wavelength is 12 centimeters. And the holes are 2 millimeters, a factor of 600.
Why do you say those two numbers are related? If that were the main determinant, the holes would be larger.
Clearly, the holes are nowhere near 12 centimeters/~4 inches wide.
So the important size is not the wavelength but the amplitude.
How much is that? I.e., what is the wavewidth of a microwave oven.
Factor of 60 (I fear I’ll never get the hang of metric).
And if they are micro, why is light smaller? Didn’t we start calculating the spectrum from an optical prism? So wouldn’t they be on the macro side of things?
I’m pretty sure that they were named ‘micro’ waves in comparison with radio waves, which are similar but even bigger.
We may have started working out the spectrum from several places at once and then figured out that they all connected. 
The holes could be larger, but then you’d have more leakage through the door. The thing about electromagnetic radiation is that it doesn’t follow nice, classical rules like butterflies and gnats do. They are governed by probability as stated by quantum electrodynamics (QED - no relation). The holes won’t suddenly block photons with a wavelength shorter than their smallest diameter, it will attenuate them - that is, they will let some fraction of them through, depending on the size of the holes in relation to the wavelength.
Adding on to what QED wrote (if I may be so bold), the tradeoff for manufacturers is between:
[ul][li]Customer appeal (“I want a window on my oven”)[]Lawsuit potential (“Your ovens leak too much microwave radiation”)[]Production cost (“Drilling smaller holes requires finer, more expensive tools”) [/ul][/li]
Given the long history of windows on conventional ovens, the customer desire to watch food as it cooks is established. It might be possible to build a screen that was sufficiently “tight” to block 99.9999999% of microwave radiation (yet allowing visible light to pass), instead of the 99.9999% blocked by a screen with 2mm holes, but it would be expensive, and the 99.9999% standard is well above federal guidelines, making lawsuits unlikely.
We’re “this close” to inboard oven cams becoming common. They’ll be linked to the TV/monitor on the fridge, and you’ll be able to see them at a URL.
-not kidding-
But I’m more afraid of color monitors broadcasting radiation than microwaves.
Guess that’s because I was around when color TV’s were forced to add more shielding in the 60’s.
So that’s what those stupid monitors on refrigerators are. I couldn’t figure out a use for them.
There’s two different types of “radiation” involved here. Radio waves (which are what microwaves are) are non-ionizing radiation, which as far as we presently know is harmless until you get enough of it to cause damage by cooking. As you go up higher in frequency, you get to light waves which are pretty much the same. You can shine a flashlight on someone without causing them any damage, but if you focus the light from the sun through a magnifying glass you can burn an ant to a crisp. At slightly higher frequencies, electromagnetic radiation starts to become ionizing, which means it can strip electrons off of atoms and create ions. This is known to cause cell damage and cancers. This starts to happen in the ultraviolet range of frequencies, which is why sun tan lotions and sunglasses make a point of being UV blockers. Cathode ray tubes, in addition to putting out visible light, also emit electromagnetic radiation well above the ultraviolet range, up into the range of frequencies we call X-rays. I’ve never measured it, but I’m told that the amount of high frequency electromagnetic radiation coming out of a modern TV or computer monitor is so low as to not be worth worrying about. If you are worried about it anyway, buy a flat panel TV. It’s only CRTs (which a flat panel doesn’t have) which emit the harmful type of radiation.