Boy, have I been in this situation more times than I can count. When you inherit furniture from your grandparents and buy a LOT of used furniture from Craigslist, well you start to get creative. I found the “musty”, moldy type smell in old furniture to be most prevalent. Washing with bleach water will kill any linger growing mold, but it unfortunately will not eliminate the odor permanently, and may also damage the wood.
Some of the favored smell-mitigators: Sunning and airing, baking soda, cat litter, and vinegar. NOT ALL TOGETHER! Read what Atta Girl found works best for getting gross smells out of furniture: