1) Dust frequently. Use a soft dry cloth (old t-shirts work well) or a gently dust with a soft duster. You may find that an every-so-slightly damp cloth helps to remove dust without stirring debris and allergens into the air.
2) Cleaning sticky spots. It's best to avoid getting solid wood furniture wet, but when the occasional sticky spot appears, it's best to use a very gentle soap and water mixture to remove the grime. Make a solution of mild detergent and water, dip your soft cloth in the water and thing almost completely wring it out so it's just barely damp, still wanting to avoid getting the furniture very wet. Rub the spot gently and then rinse with clear water, again having wrung out the cloth so it's barely wet.
3) Protecting your furniture. While some products that are specifically designed for cleaning wood surfaces do contain a silicone-based oil that may provide some protection for your furniture's finish, the best protection comes from wax. Wax leaves a nice smear-free finish and a hard layer of protection for your surfaces. Make sure to use a liquid or paste wax that is made specifically for wood furniture, and apply in thing coats to prevent a cloudy-looking surface.
It's always best to ask your furniture retailer or maker for care directions specific to your wood and furniture piece. A little TLC will go a long way in ensuring the longevity of your furniture.