• Reflect on your interview and ask yourself where things could have gone off track. Then ask for feedback. There are some useful lists here and here. 
  Go over these lists and try to be honest with yourself if any of the
  common mistakes mentioned could have been a factor. But don't assume
  that you will be able to definitively pinpoint the reason for the
  rejection by guessing on your own. Try to be reflective, rather than
  simply hard on yourself. There is a natural tendency to go negative and
  spend your time and energy kicking yourself for anything that might
  have gone wrong. Try not to do this to yourself. Everyone makes
  mistakes sometimes, and letting some voice in your head echo "stupid!
  stupid!stupid!" is no way to bolster your self-esteem. Once you have a
  few ideas, then make the call, or write the e-mail, asking if you can
  set up a time for feedback.  And definitely ask for a given time, so the hiring manager can prepare for the conversation. If they give you a specific time, they are going to tell you what they think is useful or appropriate, within whatever parameters their department or institution have set. If they don't agree to a specific time, then drop the issue and move on. No point in trying to get blood from a stone.

• Only ask for feedback if you can take criticism gracefully. 
  Even if the hiring manager shares the rationale for rejection, you may
  or may not agree with all the feedback you get. Arguing about your
  various good points, insisting you can be a fit for the position, or
  saying that the interviewer(s) misinterpreted what you said are
  sure-fire ways to seem out of touch, unwilling to accept feedback, just
  plain angry, or even worse, desperate. None of these are good ways to
  be perceived. Once the interview is over, it's over. Impressions have
  been made, and they have been set. They may not be set in stone, but likely at least in a hard clay.  If you were "almost"
  the candidate they selected, being pushy or argumentative about
  feedback will seem needy and can only hurt any chances you might have
  for later consideration.

• Commit yourself to doing something with your feedback. 
  There's no point in asking for feedback if you are not going to do
  anything with it. Once you know how you are going off track, you can
  try to fix it. Here's my method to doing something with feedback. 

• First,
  ask yourself "What part of this criticism is fair?"  This will require
  some reflection and some level of stepping into someone else's shoes.
  If the criticism is fair, then you need to make peace with it, and
  decide what you want to do with the information.

• Second,
  ask yourself "What can I do to correct the situation?" Brainstorm a
  little bit. Ask for ideas from your supervisor or a trusted colleague.
  Pick the feedback apart and hold the pieces up to the light. You will
  probably see some opportunities for growth.

• Third, take a look
  at the parts of the criticism that you believe are not fair or
  accurate. Then ask yourself why you would want to subject yourself to
  working in an environment where you will question the fairness and
  judgment of the people you work with from day one. You are better off
  for having dodged that bullet. Let it go.   

• Finally, take a look
  at any pieces of criticism that might be fair, but that you are
  unwilling to change. We all bring unique aspects of our personalities
  to work with us, and particular ideas about what we like to put out there as our public images and work personas. If the criticism was fair,
  but you are unwilling to change, then it really wasn't a good "fit" for
  you and you need to keep looking until you find the right environment,
  or the willingness to change those things that were the roadblocks in
  your way. Look in the mirror, say to yourself "I am good enough, I am
  smart enough, and one day I will find people who actually like me for
  who I am, how I am, and what I can bring to the table. It's their loss,
  not mine."

• Don't keep knocking on the door when it's been clearly closed in your face.
  Don't be the stalkery almost-lover standing outside the door saying
  "why don't you love me?" to the blind date you just met, when the date
  didn't go well. Nobody likes that. Don't let rejection wreck your
  self-confidence. Instead, let it teach you what you need to know about
  yourself and how you relate to others, so you can handle the right
  relationship when it comes along, and be ready to throw yourself in
  with all your heart. The lesson doesn't define you. The learning does. 
  Lick your wounds, dust yourself off, and stand firmly on your own two
  feet, facing the world. There will be another door. The important thing
  is to be practically and emotionally prepared to ring the doorbell and ready to walk in the next time someone answers.

If you are
one of those out there licking your wounds incurred in the brutal
realities of the job search, hang in there, and good luck!