"Let the dead bury their dead"?

Luke 9:57-62 And it came to pass, that, as they went in the way, a certain man said unto him, Lord, I will follow thee whithersoever thou goest. ⁵⁸ And Jesus said unto him, Foxes have holes, and birds of the air have nests; but the Son of man hath not where to lay his head.  ⁵⁹ And he said unto another, Follow me. But he said, Lord, suffer me first to go and bury my father. ⁶⁰ Jesus said unto him, Let the dead bury their dead: but go thou and preach the kingdom of God. ⁶¹ And another also said, Lord, I will follow thee; but let me first go bid them farewell, which are at home at my house. ⁶² And Jesus said unto him, No man, having put his hand to the plough, and looking back, is fit for the kingdom of God.  (Authorized Version/KJV) [emphasis mine]

While today's Gospel reading includes one of my favorite verses of scripture (v58), it also contains one I have always found troubling:  "Let the dead bury their dead." (v60).
I have always had difficulty trying to decipher what Christ is saying when he tells the man to let the dead bury the dead.  Is Jesus saying we should not mourn the loss of the ones we love?  If that is the case, then why is the shortest verse in the English bible ("Jesus wept." John 11:35) an exhibition of Christ's own mourning?  We must examine this statement in context.
First we have a man who wants to follow Jesus where He goes.  Christ lets him know that "the Son of man hath not where to lay his head." In other words, Christ has no home and nothing tying Him down to one place.  His is an itinerant ministry.
Then, Christ invites our guy to follow Him, but he asks if he can first go and bury his father.  And Jesus responds with our confusing verse.
And finally, another says he will follow Christ, but wants to first go tell his family goodbye.  Not an unreasonable request.  But Jesus replies with this analogy of plowing while looking backwards.
So one could assume that Jesus calls those who follow him to abandon all their earthly relationships, right?  Well not so fast.  I think more accurately, Christ is saying:  Put me first.  Do not let your relationships with family and friends, as important as they are, to become more important to you than I am.