Just use a damp, Not Wet cloth. It will remove the dust and dirt without doing harm. If the water stains are uneven tannin bleed, a wetter cloth can be used along the whole board or log to allow the tannin to run evenly. Tannins are water soluble, and will form dark rings with a lighter interior where water has dissolved them. Evenly applying moisture, re-distributes the tannin evenly. Just work one board or log at a time, and don't unevenly wet adjacent ones.

If water stains are dark in color, oxalic acid can lighten. Use this with care, you can create more problems