special relativity - Light in Different Reference Frames

I think I'm just confused, but for some reason I thought that light moving straight in one frame would have to move in the same direction in another frame. I know there are photons-but because I have not learned about them I was thinking purely from the wave model. If light has no mass, shouldn't a disturbance sent in one direction, regardless of perpendicular speed, move in that direction in all frames? In particular I had a homework problem that was asking how the Michaelson Morley experiment's null effects could be accounted for with Length contraction. They used the path of light in the vertical direction (Not a straight line.) Clearly the math works out this way, but I'm not sure why the vertically moving light would have a horizontal velocity just because the frame it was shot from had that velocity. Why does light have this inertia? Can someone explain it via wave theory?