Hose off the wicker piece, removing most of the loose contaminants and dirt. Focus on the spots that typically come in contact with clothes and body oils.
Use a horsehair brush to scrub under the wicker's weave. Rinse the piece again.
Allow it to dry naturally.
Mix the linseed oil and turpentine in a well-ventilated space (preferably outside) Rub the mixture into the piece, pressing the oil firmly into the weave. Gently dab away any excess that bubbles to the surface.
Apply a fresh coat of matching stain to the piece with the paintbrush. Allow the piece to dry for two to three days prior to use.