Do plastic cutting boards harbor more bacteria than wood boards?

For the best sanitation, you want glass, or ceramic cutting boards. (Stainless steel is also good, but hardest on knives.)

But folks tend to not like any of these because they're hard on knives. It depends on your priorities, I guess. I prefer to err on the side of avoiding food-borne infections and other grossness, and I'm willing to replace or sharpen knives.

The problem with wooden cutting boards is that they're porous, so juices (especially from meat) can seep down into the wood and provide a rich nutrient broth for bacterial growth.

The problem with plastic cutting boards is that, every time you cut on them with a knife, you make a tiny groove in the plastic, which can harbor bacteria. These bacteria growing down in a groove or fissure are difficult to wash away, even with manual scrubbing. Also, tiny bits of plastic can end up in your food.

Plastic has a slight advantage over wood in that it's easier to wash. Throwing it in the dishwasher is the easiest, but won't get it completely sanitized, especially down in all of the knife-cut grooves. Your best bet for sanitizing plastic cutting boards is soaking them in a mildly diluted bleach (sodium hypochlorite) solution. For obvious reasons, none of that is really an option with wooden cutting boards. (Small wooden cutting boards can be sanitized by heating them, e.g., in a microwave, but this doesn't work well for large "butcher block" style, which are, in my experience, the most popular kind of wooden cutting boards that are seen.)

You really never want to use a wooden cutting board for meat. For meat, use something completely non-porous that can be easily cleaned, like glass or ceramic. In a pinch, you can use plastic, but make sure to sanitize it thoroughly and regularly.

A wooden cutting board is acceptable for fruit, vegetables, cheeses, breads, etc., and many people find it easier to work with than glass or ceramic. Plastic would also be fine for these items.

If you're interested in the research, there's quite a bit of it out there, but I can't claim to have read it all. The above is a general summary, based on my background in microbiology. The classic study is by Dr. Dean Cliver of UC Davis. You can surely find links to it online.

Another study, focused on commercial kitchens and the realities of induced wear (especially with regular cleaning), was conducted by Welker et al.: "Bacterial Retention and Cleanability of Plastic and Wood Cutting Boards with Commercial Food Service Maintenance Practices", and this is freely accessible online. The conclusion (unsurprisingly) was that wood is more likely to wear out, developing "cracks […] sufficiently wide to entrap bacteria." Plastic is easier to clean than wood, and "after standard food service washing and sanitizing", wooden cutting boards promoted bacterial growth but plastic did not.

A study performed by Nese et al. ("Decontamination of Plastic and Wooden Cutting Boards for Kitchen Use"), also freely accessible online, confirms what I stated earlier about plastic being generally easier to sanitize, but the knife-cut grooves tending to interfere with cleaning. This paper also discusses the popular argument that wooden surfaces will absorb juices (bacterial inocula), thus inhibiting growth. This is true to some extent, but as other studies have shown, cannot be fully relied upon, especially as the wood ages and cracks, due to both use (knife cuts) and cleaning.

The relevant advice on the US Food and Drug Administration is, essentially, that both wood and plastic are acceptable, so long as you sanitize them frequency and replace them when they begin to become worn. (In my experience, people tend to know that plastic boards need to be replaced—not that they always do it—but they tend to assume wood lasts virtually forever, and purchase accordingly, spending lots of money on heavy, expensive wooden cutting boards.) The FDA also mentions bamboo as a possible option, since it's harder and less porous than hardwoods, meaning that it absorbs less moisture and is more resistant to scarring. I don't have experience with this, so I can't comment any further.

I leave you with a quote from Dr. Charles Gerba, a professor of microbiology at the University of Arizona, Tucson, who told Food and Wine magazine:

In most cases, it’s safer to make a salad on a toilet seat than it is to make one on a cutting board. There’re 200 times more fecal bacteria from raw meat on the average cutting board in a home than a toilet seat. Most people just rinse their cutting board, but poultry and raw meat can leave behind salmonella and campylobacter.

People disinfect their toilet seats all the time, but they don’t realize that they really need to pay attention in the kitchen too.

Indeed. In case I haven't emphasized it enough throughout this answer, the key is thorough, regular sanitation of your chosen cutting surface, no matter what it may be.