I am about to lay a hugelkultur bed and was considering using some alder (or is it birch?) which recently fell on my property.  Before it fell, it had many woody and stiff oyster shaped mushrooms all over it.  Should I be worried that this growth will negatively effect my beds?  Is there a reason to pass on that wood, and use something else instead?  I want to use it because it is already really spongy and rotten.

That is the type of wood I look for to include in my Hugelkultur.  I use younger wood, too.

Mushrooms are GOOD!

Mushrooms and other fungal growth is a sign of a healthy woodland soil culture, and a noted benefit of hugelkultur beds - speeding the succession from an immature soil ecosystem (heavily reliant on bacterial composition) to a mature woodland soil ecosystem (with a heavy fungal composition).

I love mushrooms.  My first portobellos are coming up this week in the lawn. I knew I wanted friendly mycelium, but I wasn't sure if unhealthy stuff was a good idea to encourage.

Compost made from most gardens is primarily bacterial.  Compost from the woods are primarily fungal ("mushrooms").  Combining the two is the best of both worlds.  Mushroom compost is highly sought after.