In order to go through a material evenly, light has to avoid two things:
1. being absorbed
2. being scattered off into another direction.
Whether light gets absorbed depends on largely on something that you wouldn't guess from classical physics, but is connected with quantum effects. The energy from light waves gets transferred to other things in packets ("quanta") whose size depends on the frequency of the light. The electrons in different materials have different possible states with specific values of their energies. In order for the light to get absorbed, the energy of one packet has to match up with the energy needed to boost some electrons from one state to another. It happens that in ordinary glass there are very few states with energy differences matching up with the energy quanta for visible light. Wood has lots of different compounds in it, so not surprisingly it has lots of different possible energy differences for its states, so it absorbs a wide range of frequencies.
Whether the light gets scattered depends mainly on how smooth the material is. In a material with very even composition (specifically, a very uniform index of refraction), the light waves travel straight, like waves in a shallow puddle with an even bottom. In materials whose composition varies from spot to spot, the waves scatter in different directions, like water waves in a puddle with a lumpy bottom. Wood is made of cells, with different contents in the cell walls and the cell insides. That unevenness scatters light.
You can find ordinary glass with many little bubbles in it. That sort of glass won't absorb light, but it will scatter it. I believe that the frosted glass of light bulbs just uses an uneven surface to scatter the light. Of course you can also find glass with absorbing sites in it, such as the stained glass used for decorative windows.
If you take a thin sheet of ground up wood (namely paper) it can be too thin to absorb much light yet still too scattering to see through. Usually, if a spot is soaked with oil or water, it fills in some air gaps, making the material more uniform and easier to see through.
Here's a story I hope you enjoy. A young German student, James Franck, was staring at a transparent grease spot in his Greek book, wondering if there was a connection between why he could see through it and why he could see through glass with a smooth surface but not with a rough surface. Unfortunately, he had no idea what to say when his Greek teacher asked him a question. Many years later, after Franck had conducted some of the early experiments establishing quantum mechanics, he got a letter mixed in with ones congratulating him on his Nobel Prize. It asked "Is it possible that you are the same James Franck who was so stupid in my Greek class?"
(republished on 07/29/06)