Rice can used as a low-tech desiccant for some purposes, but for long-term storage you should not rely on rice alone.
Rice can used as a low-tech desiccant for some purposes. It will dry mushrooms enough that they can be stored briefly without rotting, and will reduce their mass so they are easier to transport and consume. Table salt is another household item with desiccant properties, and it carries no risk of putrefaction. However, for long-term storage you should not rely on these materials alone. They may be used as a first step in reducing water content, like sun or fan drying, but the mushrooms should then go through an additional drying stage with a desiccant like silica gel or calcium chloride to ensure they retain maximum freshness and potency. Given the time and energy already invested at this point in the cultivation process, and the widespread commercial availability of cheap, effective, non-toxic desiccants, it is recommended that you avoid unnecessary shortcuts.