(code & PCB provided)
I used ESP32 devkit v1 as the main MCU to control three 74HC595 8-bit shift register. Two of the registers control the layer pattern (4x4), the remaining one swipe through the vertical 4 layers.
The animation affect is done by 'Persistence of vision', a frame you see is actually LEDs turning on/off rapidly layer by layer.
For series resistors I use 160 Ohms to get the brightness and yet not burning the LED.
The code is in /code/LEDcube_esp32.ino It's not pretty but gets the job done.
Here is the website that I reffered to.