When beginning a job search, you have to ask yourself two questions:
1) Academics versus private practice
2) Location, location, location
Once you have answered these questions, you are ready to start looking for a job. Some jobs are highly advertised through recruiters and gaswork. However, the best jobs out there are word of mouth. Try to develop connections through your faculty and alumni with either other academic programs or private practice groups. It is much easier for someone else to present you to a practice (i.e. faculty member) than simply cold calling.
That's not to say you can't be successful cold calling though. It takes a little confidence and practice, but you can pick up the phone and dial a department or group and pitch yourself. Most of the time the person you will be speaking to is a secretary/administrative assistant. Don't underestimate the power of this person to get your foot in the door, especially if it is the chairman's/group president's personal assistant. Ideally, the administrative assistant will give you an e-mail address or phone number of the physician you will want to contact. Also, the administrative assistant can give you a very accurate off-the-record perspective of how things are going in the department/practice like how late people really work or how much call is taken or if they are seriously in need of more anesthesiologists.
Following up can be the tricky part, particularly if a significant amount of time has passed between you giving your information and them getting back to you. Each situation can vary (a four week lapse 1 year prior to graduation is a lot different than 3 months prior to graduation). A brief, polite phone call or e-mail reaffirming your interest in the department/practice along with new information about yourself (board certification, obtaining state license, etc.) is helpful. Unfortunately, some departments/practices don't have enough class to let you know once and for all that they aren't interested. Sometimes you just have to know when to take a hint, like when the hot girl you keep hitting on persistently blows you off.
Try not to take setbacks in your job search personally. Whether or not a department/group decides to hire you is ultimately a business transaction. You can be the best anesthesiologist coming out of residency in your program/state/country but if the group you want to join is already overstaffed, then you probably aren't getting a job offer despite your awesome skills. Even if a group doesn't think you are "strong enough" to succeed in their practice, it is better to learn this prior to starting the job than a year into the partnership track.
My prediction is that the job market for anesthesiologists will continue to be a little tight as older anesthesiologists delay retirement and medicine waits to see how Obamacare will affect us all. It will be important to be proactive and persistent in your job search and to not give up.