Thinking back to my fairly limited knowledge in planetary science, having strong winds doesn't require anything as drastic as an asteroid
Winds can pick up quite easily when there is a strong temperature difference on the planet. This is mostly discussed in relation to tidally locked planets around red dwarfs where one side is permanently very hot and the opposite side is permanently very cold (unless you have a case like that of Venus with a very thick atmosphere).
As to whether it is the same kind of mechanism that starts these on Mars I am not sure. I do remember reading a few years ago now that it is also because Mars has such a thin atmosphere that allows for these global storms to begin and last so long.
I believe one of the reasons that the Earth doesn't suffer from this is because of our thicker atmosphere which helps trap and slow down heat loss. The Earth also has VERY large heat reservoirs; the oceans.