In effect, you asking about one of the basic difficulties of translation between any two languages. It's important to recognise that either style or idiom, and sometimes both, have to be transformed. In order to accomplish this, a translation is often a compromise, and the answer by Draconis indicates this very well. I would, however, be bolder and look for something which equates to the sense behind what you need to translate.
Is it unavoidable to write of something running through our blood as such? To me, it seems that if you want to refer to something within your nature — if you like, something inborn, innate — you might avoid any reference to blood and the way it moves, and still achieve what you want.
Perhaps res in nos innata, 'a thing inborn to us' would be a suitable format. You could substitute a more particular noun for res to suit your context: litterae or humanitas, for example (remembering, of course, to maintain agreement between whatever you choose and its epithet).