I seem to be getting a coating of mildew on my wood walls.

Check the humidity level. It should be about 40%, if its higher than that it can lead to mold and mildew. Scrub the walls with a bleach solution, and if your humidity is too high look into an exhaust fan or dehumidifier.

Spray the walls with bleach water. I would also run a dehumidifier.

http://homeguides.sfgate.com/clean-mold-mildew-off-painted-walls-93553.html

If it's mildew or mold you have a moisture problem. Water leak behind walls, inadequate ventilation. Cleaning is only temporary. You have to determine the source of moisture.

http://www.concrobium.com/en-can/products/mold-control-fogger/

http://www.mold-advisor.com/white-mold.html

Do you have bad windows, or somewhere humid air can get in? Maybe bad ventilation in the bathroom, etc. That's contributes to the humidity level?