Empty out the cedar chest.
Take the cedar chest to an open area such as a garage or outdoor area that is protected from the elements. Do not place the cedar chest directly on to dirt or grass if cleaning the chest outdoors. Place a tarp or blanket down to place the chest on.
Get all loose dirt particles out of the chest. Use a leaf blower to blow out large pieces of debris. Use a vacuum cleaner or shop vac to suck out the remaining dirt and dust. Do not scratch the chest when using such tools. Use soft fabric attachments for the vacuum cleaner.
Allow the cedar chest to sit with the lid open for a few hours to allow fresh air to permeate the chest. Make sure that the chest is protected from dirt, debris, odor and moisture.
Take a dry, clean dust cloth and begin wiping down the inside of the chest. Repeat as necessary until the cloth is no longer picking up dirt or dust. Use the same dusting method on the outside of the chest to clean the exterior. Pay attention to corners and crevices to ensure a thorough cleaning.
Use a fine-grain sandpaper on the inside of the chest to restore the cedar smell. Using the sandpaper, make small circular motions throughout the inside of the chest to remove a small amount of the top coat and expose the fresh cedar underneath. After completing the sanding, remove the dust from the chest.
Apply a light coat of cedar wood oil to the outside of the chest to restore the shine. Use oil sparingly and apply in a circular motion using a clean, dry cloth. Test a small, inconspicuous area of the chest prior to applying oil to the entire piece.