Let G = (Ω, E) be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Z t } on Ω whose kernel p satisfies p(x, y) > 0 ⇒ {x, y} ∈ E for every x, y ∈ Ω. In words, transitions only occur between neighboring points of the graph. Suppose further that (Ω, p, d) has coarse Ricci curvature at least 1/α in the sense of Ollivier: For all x, y ∈ Ω, it holds that W 1 (Z 1 | {Z 0 = x}, Z 1 | {Z 0 = y}) ≤ 1 − 1 α d(x, y), where W 1 denotes the Wasserstein 1-distance. In this note, we derive a transport-entropy inequality: For any measure µ on Ω, it holds that W 1 (µ, π) ≤ √(2α/(2-1/α) D(µ ll π)) , where π denotes the stationary measure of {Z t } and D(·ll·) is the relative entropy. Peres and Tetali have conjectured a stronger consequence of coarse Ricci curvature, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our approach suggests a natural attack on the MLSI conjecture.