3D Extrude & Bevel → Map Art
You can achieve this with the Map Art feature of 3D Extrude & Bevel. You can only map artwork that has been created as a symbol. This gives you the advantage of being able to create your symbols, map them to your 3D object then update your symbols and automatically update your 3D object.
Create your symbols
You need to create a separate symbol for each face of your 3D object. Since you want objects that overlap the edges you need to create your symbols accordingly:
Create your symbols by dragging the art (one face at a time) to the symbols palette.
Create the 3D Extrude
Create your 3D object (Effect → 3D → Extrude & Bevel). Use the Map Artwork... button to bring up the Map Art dialog. Toggle through each surface and select the correct symbol for that surface from the Symbol dropdown. You can transform, move, rotate & scale the symbol to fit the surface.
Final 3D Mapped Object