[bojan]

You are basically right.
What is important to note here is because moving electron carries electrical charge, when it moves this produces the change of electric field, which then generates changing magnetic field, which generates changing electric field.. etc.
Static electric or magnetic field will not generate the other one, so there will be no EM wave to propagate.

EDIT:
About the media.. since EM field is the property of space, we can safely say that the media for EM waves is space itself.