A small addition to the solution from Jonathan: use Select Mirror to quickly select the similar faces.
P.S. I used the Pie Menu addon for the Pie menus. It can be enabled here: Edit->User Preferences->Add-ons->Pie Menu
To quickly select the faces of all the corners simultaneously after you've subdivided, select the 3 faces around a corner, then:
Select -> Select Similar -> Face Regions
This will select all the additional 21 faces of the other corners in one fell swoop.
Then -to select all the insets of the cube at once, select one of the inset faces, then:
Select -> Select Similar -> Area
PS: I'm new to Blender and I'd like to know- is there a shortcut for the "Select" menu? I can see it becoming tedious having to move the mouse and actually clicking the Select menu item at the bottom everytime I need to.