Rendering Only Slices of 3D Objects

I'm curious about rendering sections of 3d objects, but not entire 3d objects. For example, in Miegakure you can shift around the 4th dimension and it looks like this https://www.youtube.com/watch?v=uWsBnVtl8tA.

I'm not interested in rendering things in 4 dimensions, but what I am interested in is how the developer slices up the objects. The whole game takes place on that small cube and I'd like to do something like that with a strictly 3d world where the player moves along and is only able to see an isometric slice of the world kinda like This image.

In researching this all I have really found so far is mesh cutting and slicing which seems really expensive and difficult to do. I wonder if there isn't some way I could force culling in the world so that certain verts and faces aren't drawn, but the objects still appear solid.

I know how I could achieve this look with a terrain object or a plane, by using a heightmap which moves around on a static terrain object, but that doesn't allow me to have other objects on the map. I'm mostly interested in a graphics trick that only renders a specific slice of the world. Any sort of direction, help, or insight would be hugely appreciated. Thanks!

 

http://forum.unity3d.com/threads/simple-cross-section-shader.34508/