A comet leaves dust behind it, so there is a constant dust stream in the same location that the earth moves through:
1) Since this dust stream is from a comet path which is constant, the earth will pass through it once a year (365 days).
So then in your model, the Sun itself does not move at all? 
Or do you believe some magical force carries this tenuous stream of dust completely undisturbed along with the Sun and its retinue of planets and their moons, as the whole system barrels through the interstellar medium at an alleged speed of 486,000 miles per hour, somehow keeping it "constant" year after year?
Ludicrous.
First off, motion is relative. The interstellar medium is very sparse and would not affect the the orbit of the comet. Anyways, the particles flying off the sun and the sun's magnetic field will overpower any interstellar "wind." The solar system's major motion is around the center of the Milky Way galaxy, which is so relatively slow that it would not affect the motion of comets and planets. Thus, for purposes of determining motion of objects in the solar system, the sun can be considered as the center, with negligible motion (except for slight wobbles mostly caused by the larger planets like Jupiter).
Given this, the comet's orbit stays pretty much the same, and it leaves behind dust and ice in this orbit, which the earth crosses once a year, causing the meteor shower.
Your turn to explain periodicity (make it better than certain stars go nova again and again every 365 days or integer multiples thereof, but never with any other periodicity, and that these stars always reach their zenith near morning. That's just desperate talk.)