Is there an easy way to create some variance in my procedural texture among objects using the same material? I created a procedural dirt texture for the rims of my tires.  Problem is every tire has the exact same pattern.  Is there any way to create some variance across the different rims?

I'm looking for the easiest method.
You can use the Random output of the Object Info node to translate the Texture Coordinates as shown :

Without translation (note how they all have the same pattern) :


With object randomized translation along the Z axis :
 

Adjust the Multiply node to vary the distribution of the randomness over the texture space (larger values will select more distant sections of the texture space).

You could similarly use the Random value to vary other aspects of each object's texture/properties.