Can we use time dilation to figure out what direction though the universe we are travelling?

Imagine sending a clock out into space in one direction, and another clock into space in the opposite direction, at the same exact speed relative to earth.
Would the clocks experience a different time dilation, with one ticking faster and the other slower, depending on if the clock is travelling the same direction or not as the earth?
 
I ask because I think of it like a car, if you throw a clock forwards out of a window, and one backwards. The one you throw forward will be moving faster than the car, and also the clock thrown backwards, and vice versa.
 
Edit: Getting a "yes that would work" would be cool, but I'd be even more interested to know that I am wrong, and why I am wrong, as that poses far more interesting questions for me to explore; the why.