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Here's an image of a cube with lots of edges added. It was created using knife tool and lots of clicking:

enter image description here

What would be the best way to randomly generate such cuts with less effort...?

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Rereading the question and answers, there is actually a third way to do this that's very very quick if you just want the surface tessellation. (i.e. you dont want the cuts to be separate objects or meshes)

Make a cube, apply subdivision surface on it (simple, 2 levels), then use the Decimate modifier to reduce the polygon count and apply the modifiers from bottom to top.

enter image description here

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The cell fracture tool should do what you require. Press SPACE and search for CELL FRACTURE

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enter image description here

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Continuing off of AdamTM's answer. Cell fracture creates a lot of separate objects. To get to just cuts on the surface takes a few more steps.

  1. Merge all of the fractured objects together Ctrl + J.
  2. In edit mode select all of the exterior faces and then use Ctrl + I to invert the selection.
  3. Use X to delete all of the interior geometry by deleting the vertices.
  4. Remove doubles to fix all of the overlapping vertexes at the outer seams.

My final result looks like this: enter image description here

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