Led Matrix Driving Method
I assume that you will be using a MCU of some description to drive the matrix. 
Your idea of using 74595 Register is a valid method however as some have said you may end up with current draw issues as most shift registers are limited to about 30ma, the SN74HC595 is limited to 35 just looking at the data sheet. 
If you have 16 LEDs driving off a row of a shift register, (for example if i I turn on a line of LEDs), only 35ma will be available to light all these LEDs. This means a very small brightness if you divide 35ma by 16 LEDs.

You may be able to get away with some fast multiplexing, only lighting up 1 LED at a time however a much more elegant solution is to use a constant current driver chip like the TLC5940. This can provide up to 130ma if I am reading the data sheet correctly. 
This should handle current much better than a shift register but you may still only end up with about 50% duty cycle  (someone feel free to correct me).

120ma / 16 LEDs = 7.5

The 5940 also supports PWM I believe giving you brightness control.