So basically you just replaced everything except for the LCD (the LCD testing is described a little further). If the LCD is not working then, this could imply that either your LCD is broken, or one of the new parts is not functioning as expected, or you have not connected things correctly.
Are you sure the firmware flash went okay? You could test the Arduino Mega 2560 by connecting it over USB to e.g. Pronterface, you do not need the LCD for testing the Arduino Mega and the RAMPS.
The LCD could be tested separately by creating a sketch in Arduino IDE and connect the correct pins of the LCD cable to some pins of an Arduino device, e.g. load the the U8Glib example sketch "GraphicsTest" and add the following constructor:
U8GLIB_ST7920_128X64_1X u8g(6, 5, 4); // SPI Com: SCK = en = 6, MOSI = rw = 5, CS = di = 4
This implies that you need to attach the correct pins on the Arduino (in my case to digital pins 6, 5 and 4 for respectively SCK, MOSI and CS to the respective EXP1 pins 5, 3 and 4):
Note that you see 5 pins attached, as the LCD needs to be powered (pin 10 on the EXP1 header) and grounded (pin 9 on the EXP1 header).
I wired this on an Arduino Uno clone and confirmed that 2 of the spare RepRapDiscount Full Graphic Smart Controller clones showed the test program.
PS, this is a simple wiring of the LCD, it displays the test in fade blue-ish, so not the bright colored white pixels, but that is normal!