As an example if you want to rig a wardrobe consisting of a cabinet and two doors using the method you are describing above rather than an armature system then each of the doors would by children of the cabinet.  To parent the doors to the cabinet select all of the doors first and then select the cabinet last.  Press Ctrl+P and set parent to object.  Now when you move the cabinet the doors will follow as they are now children of the cabinet (parent).

To set pivot points with precision you need to get an understanding of how the Origin, Pivot and Snap tools work together.  Answering your question above what you want to do is move an objects Origin, because the object Pivots around its Origin.

Origin Point Screen Shot

Using the wardrobe as an example follow these steps:

1. Parent the doors to the cabinet as described above.
2. Choose the first door and go into edit mode (edge).
3. Select the edge of the door that you want it to pivot around
4. Use the Snap Tool (Ctrl+S) and choose "Cursor to Selected".  This snaps the 3D cursor to the center of the selected edge.
5. Exit Edit mode and in object mode click on the Origin tool and select
   "Origin to 3D Cursor".
6. Set the objects Pivot point to Active Element.

Repeat the above steps with the other door.  Now when you select a door you can rotate it around the new origin point that is situated at the center point of the edge you selected in Edit mode.  When you select the cabinet and move it the doors will follow because of the parenting.

Experiment with some of the other combinations of the Snap, Origin and Pivot tools to get an understanding of how useful these tools can be for placing an objects origin exactly where you want it to go.  For example instead of selecting the edge you could select either an individual face or vertice as the "Cursor to Selected" snap point.  You could also select an edge/vert/face on a secondary object and move the 3D cursor to that point with the snap tool and then select  the original object and move its origin to the 3D cursor so your original object now pivots around the secondary object. 

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