How can I stop being ambushed by mummies?

Solution 1:

According to Shadowmathfreak's FAQ, "The mummy will scare off the first hero in line unless the Mii’s color is the same as the mummy." So to stop being ambushed by the mummy, your first hero that the mummy encounters needs to be the same colour as the mummy in question. In the case of a bright/dark/etc room with a mummy, you need to clear the room's obstruction first and the hero right after will need to be the same colour as the mummy to avoid being scared away.

Solution 2:

If a mummy is in the room, then it has a chance to scare off the first hero in line so that they don't get a turn. This occurs before the hero gets any action - including "Return" or "Team Up".

Higher-level heroes seem to be a little less susceptible to this. A level 5 hero has shrugged off the ambush, and been able to take his turn.

When you're spending Play Coins to hire heroes, and facing a mummy, you can mitigate the mummy's ambush by putting a cheap Wandering Hero up front. The Wandering Hero may be frighted, but everyone else in your party will get a chance to do some damage. But don't buy a Wandering Hero in anticipation of getting some Street Pass connections later and then leave the game - I did this and the Wandering Hero vanished.