By means of effective field theory techniques, we study the modifications of some properties of weakly coupled heavy quarkonium states propagating through a quark-gluon plasma at temperatures much smaller than the heavy quark mass, mQ. Two different cases are considered, corresponding to two different hierarchies between the typical size of the bound state, r, the binding energy, E, the temperature, T, and the screening mass, mD. The first case corresponds to the hierarchy mQ≫1/r≫T≫E≫mD, relevant for moderate temperatures, and the second one to the hierarchy mQ≫T≫1/r, mD≫E, relevant for studying the dissociation mechanism. In the first case we determine the perturbative correction to the binding energy and to the decay width of states with arbitrary angular momentum, finding that the width is a decreasing function of the velocity. A different behavior characterizes the second kinematical case, where the width of s-wave states becomes a nonmonotonic function of the velocity, increasing at moderate velocities and decreasing in the ultrarelativistic limit. We obtain a simple analytical expression of the decay width for T≫1/r≫mD≫E at moderate velocities, and we derive the s-wave spectral function for the more general case T≫1/r, mD≫E. A brief discussion of the possible experimental signatures as well as a comparison with the relevant lattice data are also presented.