This scripts is for lazy people who want to get rid of all LinkedIn messages at once. This works with the latest LinkedIn 2018. Tested on 15th April 2018

removeTheConvo.js

var removeTheConvo = function()
{
  try{
    
  $(".msg-thread__topcard-btn--icon").click();

  setTimeout(function(){ 
    $(".msg-topcard__full-width-action-button.delete").click();

    setTimeout(function(){ 
      $(".confirm-delete-btn").click();

      setTimeout(function(){ 
        removeTheConvo();

      }, 300);
    }, 300);
  }, 300);
    
  }catch(err){
  setTimeout(function(){ 
    removeTheConvo();

  }, 1000);
  }
};
removeTheConvo();

Here's an updated one - https://gist.github.com/twhitacre/d4536183c22a2f5a8c7c427df04acc90