This isn't really a wrong picture to imagine. In a way, this is what the experiments are trying to measure. You have a mirror and a laser at two points of your

This isn't really a wrong picture to imagine. In a way, this is what the experiments are trying to measure. You have a mirror and a laser at two points of your lattice (the space itself), and the "region" between them compresses and expands, which makes the laser take a slightly different time to bounce back. Far away in linear approximation (which is what we saw here), the lattice picture is quite appropriate as a first order analogy.