I had this happen to me too. I found that doing as Marty suggested is a good first step. But, I had some pieces that seemed beyond redemption. Rather than give up I tried some hydrogen peroxide, straight up, in a spray bottle to hit the areas that were affected. I let it dry and then sanded again. That made some progress. I went ahead and carved some of the pieces hoping that some of the remaining staining would get cut out, which proved to be mostly true. I went ahead and did my new staining technique, which I posted in another section of this forum. The slightly darker stain made any remaining stains disappear.
That’s not the end of the story though. I had one 10 inch plate that I wanted to use for an anniversary plate for some friends. But, it had some crazy graying in places and I knew the pattern wouldn’t get rid of all of it. So, being a big risk taker (not really), I poured a bit of straight bleach into a small cup and used a medium sized, pointed tip, watercolor brush to “paint” the bleach on the problem areas. Once it dried I gave it another sanding and was pleasantly surprised to see it pretty much gone. I just took a quick look at the mostly carved piece and I can’t tell where the stains were at. I am confident that once it is finished no one will ever know that it had so much mildew on it.
Hope that helps