Drain rods are the right way to do this and the fact that white grease is appearing on the ends of the brushes indicates you are shifting it - you just need to be more patient with it.
The way I do it is run cold water down the drain whilst rodding from the other end. The rod brushes tend to further limit the water flow whilst in there, but having shifted bits of it, once you withdraw the rods, larger bits of grease start coming out the sewer end with the water.
The cleaners that I have used are based on concentrated sulphuric acid - not caustic. The two must never come into contact with one another, and neither should be used in conjunction with drain rods because of the likelihood of skin splashes. Both causes burns when concentrated.