Bring your wood chair outside or to a well-lit workspace. Lay down a drop cloth or old newspapers to catch any solution and grime. Working in a well-lit area will allow you to see all of the little nooks and crannies, which is especially important when working on chairs with lots of intricate designs.
Use dry paper towels to wipe up any excess dirt and grime buildup. Don't concern yourself with getting into all of the grooves and corners, just wipe off the bulk of it. If you have a vacuum with a brush attachment, use it to gently brush and vacuum up more of the loose grime.
Make a solution using 1-part fresh lemon juice and 2 parts sunflower or olive oil. Combine both ingredients in a small bowl and whisk briskly for two minutes.
Dip a cotton rag into the solution and wipe the chair along the grain of the wood. Take care not to scrub the wood or saturate it with the solution, as this may damage the finish. Continue gently wiping until the grime is removed from the large surfaces on the chair. Dip an old, soft-bristled toothbrush to get into the smaller, more intricate areas of the chair --- like corners, joints and designs. Do your best to brush along the grain of the wood.
Wipe down the chair with a clean cloth and allow it to dry for about 20 to 30 minutes. When the chair is completely dry, use another clean cloth to buff it back to a nice shine.