Q:

Is there a national death registry? 

I believe at least 2-3 of my family branches ended in deaths that were not handled by family (meaning they died alone or no one bothered telling anyone), so the only record of what happened to them might be a death certificate.  
Given that I don't know exactly which state they might have died in, and it is not feasible for me to contact every state, and every county looking, I am wondering if there is a centralized national database that contains death records that you can just search for the name? 

A:

The Social Security Administration's The History of the Social Security Number gives a timeline for the creation of the SSA and shows how the use of the Social Security number (SSN) as an ID number has expanded over time.   
It is useful to search more than one online source for the Social Security Death Index because not all sites report exactly the same information and they display it in different ways.   
There will be cases where you find individuals with the same name, and/or your relatives' names may not be what you expect.   Indirect evidence can help when sorting out which one of these index records might be your relative.   If they were not in touch with the family, there might not be obituaries for them, but if they had siblings, examining the obituaries of all of the siblings might help you narrow down when everyone had lost touch.   Death notices may be very brief and not give much identifying information.   City directories or telephone books might give clues to their residence before death; city directories and employee newsletters might help fill in the gaps.   If a person is well-known in their company, you might find a notice of their retirement in the newspaper or in an employee newsletter.   
Write biographical profiles and then work your way forward in time in small steps, working from what you know.   It isn't always possible to make a large leap from the last known residence to the records that turn up in the SSDI.   Narrowing down the timeline and possible locations will save you having to order several different SS-5s (application for the SSN) simply to find out if the search result is really the person you are looking for. 

Edited to add: for an important update about what the public can access, see this post by The Legal Genealogist (Posted on December 30, 2013 by Judy G. Russell):
SSDI access now limited 

Here’s the big immediate impact, and it’s from the fact that the FOIA
  exemption took effect the minute the budget bill was signed:
  genealogists should not — I repeat, we should not — order SS-5 forms
  (requests for issuance of a Social Security number) for anyone who has
  died in the last three years. 
The FOIA exemption means that all requests for “information on the
  name, social security account number, date of birth, and date of death
  of deceased individuals maintained by the Commissioner of Social
  Security” of people who’ve died in the three calendar years before the
  request will be denied.  So there’s no sense in wasting our time or
  money sending in tons of requests for SS-5s on recently-deceased
  relatives. 

A:

As far as I know the only approaching a national death index for the US is the Social Security Death Index which is described by Wikipedia as follows:

Most persons who have died since 1936 who had a Social Security Number
  (SSN) and whose death has been reported to the Social Security
  Administration are listed in the SSDI. 

So while it won't list everybody it's probably a good starting point - if the name is common it can help to narrow down the likely candidates if you know what state the SSN is likely to have been issued in. 
Most of the big sites like Ancestry will include the SSDI in their databases and there is also a list of sites where you can search it at the end of the Wikipedia article.