Do meteorites streak across the sky in the same direction?
No, meteorites of the same shower look as coming from the same point in the sky. They are usually named according to the point they seen to come from, e.g. the leonids, or the perseids. The point, they seem to originate is a result of the relative velocity of the meteorites to the earth.

Meteor showers are thought to be remnants of comets, hence the similar orbits of the meteorites within a shower.

Here a more complete list of meteor showers.