Here's a few I've noticed from the forums and my own poking around in the tutorial:
While most mythological units and tokens require incense, diamond, and other specialised resources, a few only require very basic resources. Singasteinn, for example, only requires bacon (but lots of it).
If you need to swing the course of a battle, using direct intervention to upgrade a unit or two is often a less drastic move than dropping a mythical beast into the middle of the fight. Plus it doesn't cost any AP.
If one faction is falling behind the other, you can use the stronger faction's turns to help out the weaker one by placing useful terrain (especially Grassy Ruins), or - in more extreme cases - mythological tokens that either side can use.