The words you use to describe the parts of the display and which sketch you are using need to be more precise.
Regarding the part you linked to in your question, the dark blue part is the LCD (Liquid Crystal Display). Not many buy just this part. It is too hard to use.
Under it is a green board with the commonly found Hitachi LCD Controller. This is what most people buy and call the LCD. Many Arduino sketches are written for the Hitachi LCD Controller. While you could connect this type of board directly to an Arduno, you would have to make a special cable which must follow the pin assignments in the particular sketch you are using!
Finally, the darker board beneath the green board appears to be an Arduino shield. Normally Arduino shields adapt what ever circuit happens to be upon the shield to the Arduino Uno pins and form-factor. However, again, you mush use a sketch which which follow the pin assignments for the particular shield!
Armed with this knowledge, you should be using/looking-for a sketch specific to your Arduino shield. I suspect you are using the wrong sketch and this is resulting in "white blocks" on your LCD display.
It is possible your contrast in incorrectly set. Identify if this control is available on the shield you are using and consider adjusting it.
Is the shield plugged in correctly? Check to make sure all the pins are connected to the Arduino. Most shields are designed specifically for the Arduino Uno form factor.
If you are still having problems, consider buying everything (the Arduino Uno and the LCD shield) from one specific well known vendor which can supply support through direct contact or forums. Usually a good vendor will provide support especially if they provide all parts of a project.
You should find a sketch which works with your LCD & Buttons Arduino shield. However, as you have figured out the paths between the Arduino and the Hitachi like LCD Module you might try using the LiquidCrystal Library. When this library is instantiated, the pins of the Arduino which are to control the Hitachi like LCD module are passed as arguments. Using the information in the question, it is likely the instantiation should look similar to this:
LiquidCrystal(12, 10, 5, d5, 4, 3)
...Note, you are missing "d5" in your question. So the above is likely not correct. Also, from your description, the Arduino appears to control contrast ("V0" pin) for this shield. As the LiquidCrystal library does not appear to support this, you may need to create additional code to control contrast.