| I have a British book which pictures 3, Trametes, Polystictus, and Daedalea. You might want to Google all 3 names to see which you have, although the author's instructions appear the same. They might harbor insects so she suggests using half water and half methylated spirits (would this be our rubbing alcohol or peroxide?) and soaking fungus or just brushing with paint brush. Mentions that she sometimes doesn't take this action as might lessen the effect of velvety surface. Your call...my method would be to seal up in plastic bag for a day or two ...maybe with opened container of alcohol... to see who climbed out to investigate. lol I've had only small ones and just sealed up in silica gel...figured it dried out insects too.
Fungi will dry naturally on absorbent paper in dry place. Can be put in oven at 200 F but cautions that may shrink and curl a bit. Once hard and dry, by either method, they will keep indefinitely.
Polyporus is best dried in Borax for up to 3 weeks as it holds more moisture and otherwise may mold before completely hard-dried. I must give full credit to the very helpful author, Maureen Foster, for the book on which above info is based: "The Flower Arranger's Encyclopedia of Preserving and Drying". It was printed in 1988 so may be out of print but worth searching out...one of my favorites. Let us kow how it comes out..josh | 
