Is it safe to eat cooked shiitake steams?

People always say not to eat stems because they are "woody." I regularly enjoy portobello stems (which people also say not to eat) both raw and cooked. Is there any real reason not to eat the stems of shiitake if you're using them fresh? They have so much flavor in them. I know they can be cooked in stock and I sauteed them with the caps to add more flavor but I really don't want to just throw them out, it seems like a waste.