We investigate the number of samples required for testing the monotonicity of a distribution with respect to an arbitrary underlying partially ordered set. Our first result is a nearly linear lower bound for the sample complexity of testing monotonicity with respect to the poset consisting of a directed perfect matching. This is the first nearly linear lower bound known for a natural non-symmetric property of distributions. Testing monotonicity with respect to the matching reduces to testing monotonicity with respect to various other natural posets, showing corresponding lower bounds for these posets also. Next, we show that whenever a poset has a linear-sized matching in the transitive closure of its Hasse digraph, testing monotonicity with respect to it requires $\Omega(\sqrt{n})$ samples. Previous such lower bounds applied only to the total order. We also give upper bounds to the sample complexity in terms of the chain decomposition of the poset. Our results simplify the known tester for the two dimensional grid and give the first sublinear bounds for higher dimensional grids and the Boolean cube.