In this paper, we present a new technique to generate unbiased samples on isosurfaces. An isosurface, F(x; y; z) = c, of a function, F, is implicitly defined by trilinear interpolation of background grid points. The key idea of our approach is that of treating the isosurface within a grid cell as a graph (height) function in one of the three coordinate axis directions, restricted to where the slope is not too high, and integrating / sampling from each of these three. We use this unbiased sampling algorithm for applications in Monte Carlo integration, Poisson-disk sampling, and isosurface meshing.