The general solution is just as dadt says. The idea is that the shadow is whatever color there is from ambient lighting when your directional light is *not* shining on it. So set up an image based light with the general color you want for things that are not lit by your directional light.

A specific shadow color on a particular object can be done in the material room for the object that is showing the shadow. I have demonstrated this in the image above.

I use a Blender node to pick two colors for the ground. I drive it with the Diffuse node. The Diffuse node all by itself can detect where shadows fall. By inverting it with the subtract node, I get a high value where there is shadow. I can then adjust the Blending value to decide how much to change the color. Thus, I can choose a color for the shadow.

In this case, I made a pink ball and made the ground shadow pink as well. This is nice as it gives a tone to the ground making it seem like there is some light bouncing off the ball onto the ground, a really cheap form of cheating global illumination. It's not a general solution, but can be used for special effects quite well.

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