Try mineral spirits....AKA paint thinner.
Is it painted or a natural finish? If it's paint you might be removing some of the paint along with the grime.
Mineral spirits is a lot more gentle on existing finish than Windex.
Linda C

"Fantastic" cleanser is great for grime. Be very gentle, if it is one of those with a decorative filligree'd paint job.
a coat of car wax will be just the thing to protect it once it's clean.
Casey

"Fantastic" will strip a lacquer finish.

The finish may be varnish and this can soften and become tacky in damp, warm weather. The other possibility is the 'dirt' may be condensed kitchen grease. Cabinetry and furniture in the vicinity of kitchens in old houses often became coated with grease from cooking activity.

Vinegar works great on cleaning old finishes