Our relationship to the Greeks is the same as the Greeks' relationship to the other cultures I mentioned. We stand on their shoulders and they stand on the shoulders of those who came before them. They are only a great influence on us because we gained the knowledge from them first. If all the knowledge of Rome and Greece had been completely lost after the fall of Rome, we would be attributing great influence to the Arabians, who themselves benefited from the Greeks, Romans, etc.
The factor that makes cultures "smart" is a value system that holds knowledge and learning in high regard. It's not a factor of genetics, race, or ethnicity. Anyone can be "smart" if they value knowledge. No one is born that way. You have to go through life accumulating knowledge from others, or by adding to it through experience and experimentation. In order to be particularly intelligent, you have to interact with a large variety of different people and their stores of knowledge, and you have to have the free time, finances, and desire to have experiences and devise experiments to increase your own knowledge independently. This is what the Greeks did, or at least the best of the Greeks who we remember today. All societies who have left a legacy of science have done this as well. We can do this too; any of us who have the right value system. Unfortunately, the majority of any given society lacks the time, finances, or values to pursue knowledge. Which is why the peasant-minded people you may run into at the mall won't find Greece on a map.
To say that someone or some group is smart because they are born that way is patently false.