With 3 HC595 you are not multiplexing, just sending out 3 bytes of data.
Connect SCK to SRCLK of each device.
Connect SS to RCLK of each device.
Connect MOSI to SER In of first device, Q7 of 2nd to Ser In of 3rd.
OE/ of all three to a PWM pin for brightness control, or to Gnd.
MRCLR of all 3 to +5.
Connect 0.1uF cap from Vcc pin of each to Gnd.
Don't forget current limit resistor for EACH LED, sized appropriately:
(5V - Vf)/.02 = lowest resistor value, higher is ok.
May have to tweak value so that perceived brightness between colors is the same.