This is how I cleaned up and removed mildew that grew on a wood floor.
I left a mattress directly on a hardwood floor, and some mold grew underneath. There is no insulation in the crawlspace beneath the floor, so the temperature differential between warm bodies sleeping on the mattress and cold air under the floor resulted in condensation between the mattress and the floor, which is a perfect place for mold to grow.
Mildew can also grow under potted plants and carpet, against uninsulated walls, in damp basements, if you have a water leak, or even under wet laundry left sitting too long.
Step 1: Try Wiping Up the Mold
If the hardwood floor is properly sealed, you can just wipe up the mildew with vinegar, soap and water, dilute bleach, or some other common cleaning solution. In our case, most of the mold cleaned up easily, but some had gotten into the grain of the wood. The grain tends to suck up anything liquid, including the polyurethane used to finish our wood floors, which left those areas imperfectly sealed and able to absorb moisture. In these areas, the mold was able to grow a few millimeters into the wood, making it impossible to wipe away.
The mildew growing on the intact polyurethaned surfaces wiped up so quickly that I didn't get a picture of it before we cleaned it up.
Step 2: Sand the Floor
Using 100 grit followed by 220 grit, I sanded all the areas with mold. Since I didn't want to aerosolize a bunch of live mildew spores, I periodically sprayed a dilute solution of bleach on the areas I was sanding.
In the images, you can see that I tried removing the mold with steel wool. That was unsuccessful, so I sanded. If you use steel wool, be sure to clean all steel fibers up as any left may rust.
Step 3: Remove Dust
Since the polyurethane should be applied when the wood is dry, I removed the sawdust by dry wiping with microfiber followed by vacuuming.