Chyr has confirmed on Twitter that the description for this achievement is currently not accurate. The correct description should be "Complete the Game Without Placing a Single God Cube." December update: it looks like the description got fixed!
The cubes that make you enter the dark world when you pick them up? You'll have to beat the game without placing any of them onto their respective trees.Here is a video of what the run looks like (not mine).
Along the way, you should also unlock the Homecoming achievement as you'll almost exclusively make use of hidden paths to get through the world.