Time-dilation due to relative velocity states that:
From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The second postulate of special relativity states that the speed of light is constant in all frames, which implies a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running more slowly.
Source - http://en.wikipedia.org/wiki/Time_dilation
I disagree that the light pulse traces out a longer, angled path. Rather I propose that the length of the path is exactly the same, and the discrepancy lies instead at the point on the mirror where the light pulse hits.
Imagine two mirrors A and B, a light pulse is emitted from the far right of mirror A toward B, however as B is moving at near light speed, by the time the light pulse reaches B it hits the mirror in the middle, and reflects back toward A. Again by the time the pulse reaches A, it has continued to move and so the pulse hits at the far left of the mirror.
Another example is as follows:
Imagine having a photon gun that could shoot individual photons, a friend located a distance away also had a gun that could fire single particles close to the speed of light, now, I called 'pull' and he let his particle fly, that particle/clay pigeon is travelling at near the speed of light, meanwhile I am at a right angle from it, and so I would have to fire my photon in front of my friends shot in order to intercept it, as light takes time to traverse space/reach the pigeon.
I do not view the light as travelling sideways and hence tracing out a longer, angled path rather I view the mirrors as travelling sideways, and lets face it at a speed very close to the speed of light, resulting in the pulse hitting at a different place on the mirror, and hence mainting its speed, and distance travelled, nullifying any need to introduce time-dilation.