One way we would suggest you try is oxygen bleach. There are a number of powdered oxygen bleach products on the market, and chances are you already have one in your home. Oxygen bleach, unlike Clorox bleach, combines natural soda ash or natural borax and hydrogen peroxide to form a powdered substance. When this substance dissolves in water, oxygen is released to work on tough stains and ground-in dirt. Oxygen bleaches are completely environmentally friendly and they won't compromise the color of your wood deck's stain.
Common oxygen bleach products include Oxiclean, Oxy-Boost, Ajax, Wolman Deck and Siding Brightener and Clorox Oxy Magic. To use the cleaner on your deck, simply mix the powder with the recommended amount of warm water, and work in sections across your deck, wetting the deck, allowing the product to sink in, then scrubbing the section with a long-handled brush before rinsing the area clean.
If you don't have an oxygen bleach product readily on hand, check your laundry detergent to see if it's made with sodium percarbonate or sodium perborate. These are the active ingredients in oxygen bleach products, and you can simply mix your laundry detergent with warm water to clean your deck in a similar fashion.