When teak furniture gets dirty, wash it by hand with soapy water (dishwashing detergent is fine). Rinse afterward. Hand-washing cleans the wood and helps get rid of moss. Don't use a power washer--that can damage the surface.
Use a water-based teak cleaner if your furniture is really grimy or stained. It removes deep-seated dirt, mold and mildew and makes the wood look fresher and brighter.
If you want to keep the wood from staining, apply a water-and-stain guard each season (see Resources below). However, don't use an oil sealer on outdoor teak furniture--it leads to discoloration due to mold.
Leave your teak alone if you want it to weather naturally to a silvery tone. If you love the color of new teak, you can buy a water-based sealer that contains protection from ultraviolet rays (see Resources below). Apply it once a year or more, depending on your weather conditions.