(I think) I finished routing a board, using SMD packages. Everything fits under the display.
Not really sure that all the connections are good, as I don't have the display here and therefore cannot test pinspacing and such.
PDF : http://melka.one.free.fr/temp/74HC595_RGB_MATRIX_DRIVER_BOARD.pdf
EAGLE : http://melka.one.free.fr/temp/74HC595_RGB_MATRIX_DRIVER_BOARD.zip
Cheers