Each mushroom species has its own favorites. You can find recommended types of wood under each mushroom species when looking at mycelium plugs, fast growing mycelium or log growing kits.
Generally speaking, most deciduous trees do well.
Harder types of wood such as oak and alder produce fungus for a longer time as the mycelium has more compact nutrients to consume.
Oak and birch are two types of wood that work for most of the species we sell. But as I said, take a look at the species you intend to graft and check the specific recommended wood species for that particular species.