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Wipe the furniture down with a damp lint-free cloth, following the direction of the wood grain for regular cleaning. Wipe the veneer dry with a lint-free cloth, again following the direction of the grain. This helps prevent snags that may cause additional damage to the veneer.
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Add a few drops of oil soap to a small bowl of warm water. Swish the liquid around with a lint-free cloth to blend the soap in.
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Wring out the cloth and use it to wipe down spills or caked-on debris, following the veneer grain. Follow up with a damp lint-free cloth and then a dry cloth, always wiping in the direction of the grain. Use the oil soap mixture when going after fingerprints, dirt and grime buildup.