I use one of the Park Tool thingies with a detergent base. I also use Squirt wax lube on my chains now. It doesn't take much to get the chain clean when using wax. But even when I used a teflon dry lube I would do this and it would get clean. I'd rinse with water, run it through a dry rag, let it air dry too and then reapply lube. The nice thing about using a wax based lube is that it doesn't get very dirty in the first place. The dirt may stick to the wax but it then falls off together. My trainer mat is always covered with waxy crud in the fall when I first start riding indoors.