Supposedly, wooden cutting boards should not be submerged in lots of soap and water because too much moisture leads to rot. How can one optimize for cleanliness while also caring for the board?
Mineral oil is your friend
As you said, Ana, the biggest enemy of a clean — and warp-free — wood cutting board is moisture penetration. The first trick to fight this is to wipe down your cutting board fairly often with mineral oil or other food-safe wood treatment.
This helps with cleaning as it prevents liquids and bacteria from penetrating into the wood. It also has the nice effect of helping to keep flavors from embedding and transferring to other foods.
Scrape it. Scrape it good.
The second trick is to use a sturdy steel scraper to clean the surface as you go, scraping out excess moisture after each round of cutting before it can penetrate too deeply into the wood. This is something you'll especially want to do with an end-grain cutting block, which is much more absorbent than other grains.
Soap & water is (mostly) all you need
When I'm done, I usually wash with light soap & hot water on the board, followed by a good hard scrape with the steel scraper, so I have a nice dry, fresh surface for the next use. And store that board on its edge, not flat! This promotes even drying and prevents warping.
The smart people at America's Test Kitchen used science n' stuff and found that soap & hot water was pretty effective at killing bacteria, though the USDA recommends an application of bleach dilute to really sanitize that bad boy.
Maintained board = clean board
Notable board maker, Boos, has a good list of board care suggestions, including these related to cleaning:
Periodically (once every 3-4 weeks, depending upon the use and household conditions), apply an even coat of Boos ® Mystery Oil and Boos Block ® Board Cream to all surfaces of your wood cutting board. The Boos ® Mystery Oil acts as a shampoo that contains oils and minerals that penetrates through the wood surface to increase its longevity. Next, you will want to use the Boos Block ® Board Cream, that acts as the conditioner, to seal the top coating of the wood surface.
DO NOT allow moisture of any type to stand on the cutting board for long periods of time. Don't let fresh, wet meats lay on the board longer than necessary. Brine, water and blood contain much moisture, which soaks into the wood, causing the cutting board to expand, the wood to soften, and affects the strength, of the glued joints.
Use a good steel scraper or spatula several times a day, as necessary, to keep the cutting surface clean and sanitary. Scraping the surfaces will remove 75% of the moisture. Do not use a steel brush on the cutting surface of your board.
DO NOT cut fish or fowl on the work surface of your cutting board, unless you have thoroughly followed the instructions in step #1. The moisture barrier must be intact prior to cutting any type of fish, seafood, or fowl on the work surface of your cutting board. ALWAYS CLEAN THE CUTTING BOARD SURFACE THOROUGHLY AFTER CUTTING FISH OR FOWL ON THE WORK SURFACE.
Sanitize cutting board by wiping all surfaces down with mild dish soap and water. Dry throughly. DO NOT wash your wood cutting board with harsh detergents of any type. DO NOT wash your butcher's tools on your cutting board surface. DO NOT put Boos Block ® wood cutting boards into dishwasher.
I usually scrub mine clean with a stiff brush, rinse it off, dry it off, and then lightly spray it with a vinegar solution and let it dry in a well ventilated place. I also keep my board oiled, which helps with all this.
More importantly, I don't submerge it in hot water and soap because it may cause the glue holding the boards together to come apart (which has happened to me).
Nobody has mentioned the "traditional" methods of using salt (alone) or salt with lemon juice.
After scraping off bits and crumbs, rinse off under a tap, scrubbing with a hard bristled brush if necessary, for example if stained by food like parsley, red cabbage, etc. Dry off excess water with a paper towel, then whilst still damp sprinkle salt over the surface and use a quarter (half for large boards) lemon to "scrub" the salt well into the surface of the chopping board. (You can choose whether to leave it a few minutes at this point or not). If not using lemon, sprinkle salt onto the wet board and scrub in with a stiff brush. Rinse off the salt thoroughly and pat dry with a clean paper towel. Leave upright to dry quickly in an airy place.
Food-safe oil (commercial product) can be used to "feed" the wood and prevent the wood drying out.
I prefer the vinegar solution as rfusca mentioned, I actually keep a spray bottle of white vinegar for general house cleaning. But you can also use a bleach/water solution as well. You soak it in a bleach/water solution for 2 minutes then rinse it off with a lot of water. I know some people who only do the bleach once or twice a year for a really deep clean but I've seen some people in the food industry do it every time they use their board, too.
When you dry it make sure you use a hook or dish rack so it can dry on both sides.