Many newspaper articles have said that the Higgs field acts like a "cosmic molasses". That's a terrible analogy. Molasses causes friction, which makes things come to rest with respect to the molasses. Yet, as you noted, the Higgs field lets things keep moving freely.
The Higgs field, like all other known basic ingredients of the world, has relativistic symmetry. That means that it doesn't single out one state of motion over any other. What it does have is some interaction energy with most particles, so that even from the point of view that says that a particle is at rest, it already has some energy, or rest mass.
Mike W.
(published on 07/07/2012)