This is a topic that's been covered before; I know Ashley has written up some tips for surviving the rainy season and using bamboo charcoal. So those links are worth checking out. For some additional ideas, here are the basics of what the Japanese article suggests:
1. Use newspapers, unglazed pottery, or washi (Japanese paper)
Newspaper is something I was aware of and have tried as a water-absorbing device. Scrunching up old newspaper and stuffing it into wet shoes is a surprisingly effective way to dry out damp footwear (Yoshie was skeptical about this one when she visited me once, but was pleasantly surprised when I turned out to be right). Apparently lining the floor of your closet or the bottom of your drawers with newspaper can aid in dehumidification of your home.
Using washi or shoji paper in a similar fashion also helps absorb humidity. This helps explain why they were used in older homes in Japan. Though they are pretty terrible for insolation, the papers serve to absorb humidity, which would have helped at a time when there were no air conditioners and ventilation and absorption were probably the most effective ways of dealing with the heat.
Unglazed pottery is something that wouldn't have occurred to me, but I guess it also absorbs moisture from the air, so having some pots around for decoration could serve a dual purpose.
2. Charcoal, and not just the "good stuff"
|The "good" stuff.|
If you search around, you can find activated charcoal or bamboo charcoal for sale. These work well not only as deodorizers, but as moisture absorbers, as well. If you ever use charcoal for camping or BBQing, you can recycle that as your own (non-store-bought!) activated charoal. Some people use these charcoal "remains" as fertilizer, but if you want to, you can rinse them off and redry them in the sun, and then use them to dehumidify your house. You'll probably want to keep the charcoal on a piece of cardboard or paper, or better yet in some kind of open container that you can cover with a paper "lid."
3. Baking soda isn't just for deodorizing
If you want something a little more fancy (both in appearance and effectiveness), you can pour the baking soda into an unglazed pot. Fill it about halfway, and if you like you can even tie a bow or doily around it to be extra fancy. The accessory is not recommended for guys as it can compromise your maleness.
There you have it. If you have any other ways of dealing with the humidity, let us know. Doesn't even have to be "eco!"