Wait for a sunny, warm day with low humidity. Move the piece of furniture outside, if possible. Lay down one drop cloth to keep it from getting dirty. Lay down a second drop cloth, and put all cushions too large to launder flat on the sheet.
Brush the furniture using a clean, dry scrub brush to remove surface mold. Allow the furniture to sit in the sun for several hours to kill off mold spores.
Vacuum the furniture and cushions thoroughly before moving them inside. Use a brush attachment to simultaneously suck up spores and draw out spores sitting deeper in the cushioning.
Apply aerosol fungicide, available in hardware stores, to the furniture. Allow the furniture to dry, and move it back indoors.
Smell the furniture for mold. Repeat the process the next day if the odor remains, but this time apply a solution of denatured alcohol and water first to kill remaining mold. Mix denatured alcohol and water together in equal parts, dampen and wring out a cotton cloth in the solution and rub it over the furniture. Let the piece dry in the sun, and reapply the fungicide.
Run a dehumidifier for one or two days in the same room as the furniture to suck moisture out of the air and to inhibit mold growth or regrowth. Protect the furniture from moisture from cooking, the bathroom, the laundry and other likely sources to prevent future outbreaks of mold.