If an infinite Euclidean graph is sufficiently regular, then random walk converges to Brownian motion. To what extent does this hold for irregular graphs, e.g., random LQG-like graphs? I will discuss joint work with Ewain Gwynne in which we identify a family of graphs upon which the trace of random walk, with a canonical choice of conductances, converges to Brownian motion. Our assumptions are deterministic and mild and the result applies to Delaunay triangulations with vertices sampled from a d-dimensional GMC measure.