Shake off the dust

Jesus instructed the 12 Apostles on how to respond to people when he sent them out:  

"Now whatever city or town you enter, inquire who in it is worthy, and stay there till you go out. 12 And when you go into a household, greet it. 13 If the household is worthy, let your peace come upon it. But if it is not worthy, let your peace return to you. 14 And whoever will not receive you nor hear your words, when you depart from that house or city, shake off the dust from your feet." 

Matthew 10:11-14

I think the shaking of dust off the feet or garments by the messenger signified that they were now clean from those they had warned: they would no longer be held accountable for those individuals.  

Another figure of speech connected with this is "your blood is on your own heads."  We see this concerning the Jews who rejected the message of the Apostle Paul that Jesus was the Christ: "But when they opposed him and blasphemed, he shook his garments and said to them, 'Your blood be upon your own heads; I am clean. From now on I will go to the Gentiles.'" Acts 18:6   

He was in effect saying to them: "I told you and you did not believe me, so I'm no longer responsible for what happens to you; I am clean of you."