RussT, here's a handy picture to help you understand why dark matter doesn't always follow a straight line, and why dark matter can have transverse velocity:

Now, imagine that that particle is a particle of dark matter. I've shown its velocity, and have broken that down into radial and transverse components. The radial component is simply the component of the velocity which follows a line radially inward or outward from the sun. The transverse component is whatever is left after the radial component is taken (and is always orthogonal to the radial component).
The green line in the picture is the path that that particle takes. The reason it takes this path is because it is gravitationally attracted to the Sun in the middle, and thus is accelerated toward it. But because it has some transverse component to its velocity, it doesn't follow a straight line into the sun. Instead, it follows the hyperbolic trajectory you see there.