Hey Alex,
You will still need 16(8 col + 8 row) wires for every plane because the 7219 is sort of using it as a grid where it addresses each pixel individually via a column and a row wire. If you tie any planes together then they would show the same data.
I'm thinking that with your design you bring 1 row wire down at each led. Use solid tinned wire with no insulation so that it is not so noticable. This leaves your planes totally separate and all your connections in the base. Just like the cube kit that i posted about earlier.
Make sure that the wire is bent out a bit so it doesn't short against the other row wires.
http://www.hypnocube.com/Kit2.php See the bottom image on that page. it has complete instructions on physically building the led cube.
mike