Rendering Slices of A Scene?

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(http://www.blendernation.com/wp-content/uploads/2014/02/isometric-rendering-tutorial.png) 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!