I used a similar method to Daz. I acquired some grimy pieces so I washed them in my kitchen sink. First:
I disassembled all pieces (even mini-fig arms and hands).Second:
I put the plug in the kitchen sink and added a rubber mat (the rubber mat is intended to help protect items when washing in the sink... it has lots of small holes which keeps elements from sneaking into the drain.Third:
I half fill the sink with slightly warm water and a small amount of dish soap.Fourth:
I put about 100 pieces at a time in the water and let them soak for 10-20 minutes.Fifth:
I hand wash each piece using my fingertips or an old toothbrush.Sixth:
With another rubber mat in the other side of the sink, I individually rinse each piece until they look clean and all the soap is gone.Seventh:
I use a towel to lightly pat dry the rinsed piece and place it on a laid out bath towel.Eighth:
I let the pieces air dry on the towel, or put them in a plastic container that I can let air dry in a place where I don't have to worry about the box being upended.
I repeat this process until everything is clean. It takes a long time, but all pieces are super clean afterword and they have the bonus of being super "grippy". Oil free bricks stick together better than anything. If it wasn't so much work, I'd be tempted to wash all my pieces.
I also used this method to remove tobacco smoke odor. I bought a used set from a smoking household and the pieces reeked of smoke. After the wash, they had no smell.