Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?

What methods are available for proving such a property for some family of polytopes?

Remember, the Ehrhart polynomial $p(k)$ for a convex polytope $P$ with integer vertices is given by the property that $p(k)$ counts the number of integer lattice points in the $k$-dilation of $P$, where $k$ is a positive integer.