Many of Jesus' teachings seem difficult at best, and sometimes near impossible. How are we supposed to just stand there and let people hit us? It seems cowardly or just plain stupid. What is the point? It is a bit complicated and certainly counter cultural for us. However, when you come to see this as an example of passive resistance it takes on a whole different meaning. It is rather an act of defiance in the face of power, an act of courage in the face of danger, and an act of faith that God's love and justice can prevail.
Look more closely at the text. If someone strikes you on the right cheek, turn to him the other also. Matthew 5:39.
Take into consideration most people are right handed. A strike to the right cheek is not a punch with the left fist. It is instead a slap with the back of the right hand. It is what happens when a person in power attacks someone of lesser station. Also it is the sign of an insult.
With this in mind think again of Jesus saying. If we did something to deserve the punishment or insult, we should receive it willingly and willingly receive more. On the other hand if we are innocent as we ought to be or simply the weaker person in the face of injustice something else will happen.
Take into consideration the Middle Eastern context. In that culture having and saving face was and is today important. If when someone unjustly slaps you in the face and you respond in anger and with aggression you look bad, even if innocent. If you escalate from slap to slug yourself, you not only look bad, you invite an escalation from the other and are now in even more trouble. However, if you do as Jesus says and turn the other cheek you have just taken the upper hand.
Now if your attacker escalates he is likely to punch with the right hand. You suffer the temporary black eye, he suffers the loss of face, which in that culture was a much worse wound.
When this teaching was applied by the followers of Ghandi, and Martin Luther King Jr. non aggressive passive resistance drove the British from India and won the Civil Rights movement.
Turning the other cheek was not and is not an act of weakness nor cowardice. It is a powerful strategy for overcoming injustice.