Physicists have long wondered whether the gravitational interactions between matter and antimatter might be different from those between matter and itself. Although there are many indirect indications that no such differences exist and that the weak equivalence principle holds, there have been no direct, free-fall style, experimental tests of gravity on antimatter. Here we describe a novel direct test methodology; we search for a propensity for antihydrogen atoms to fall downward when released from the ALPHA antihydrogen trap. In the absence of systematic errors, we can reject ratios of the gravitational to inertial mass of antihydrogen >75 at a statistical significance level of 5%; worst-case systematic errors increase the minimum rejection ratio to 110. A similar search places somewhat tighter bounds on a negative gravitational mass, that is, on antigravity. This methodology, coupled with ongoing experimental improvements, should allow us to bound the ratio within the more interesting near equivalence regime.