Assuming you plan on multiplexing to control each level
For a 3x3x3 cube, you will have 3 levels of 3x3 leds. For each row in a level you connect either all the common anode or cathodes together (depends on what you're using). To light that level, you source or sink current to the 3 wires that each run through 3 leds (3x3=9 leds or one level)
Lame text summary:
Source or sink (common anode or cathode leds) current to the "W"ire above to light the level.
The next level, move over where you drop the current wire to your breadboard / through your bread board to keep things neat and tidy: