So for those who are getting started, here's the workflow. It's really very simple and surprisingly effective:
I start off at stevemorse.org -- Steve is a brilliant man and a mensch, and he has a whole bunch of pages on his web site that are user interfaces onto various genealogical databases that are out there. In particular, I make extensive use of his links to the New York City birth, marriage, and death records, which the Italian Genealogical Group in New York volunteered to index. So the search form is at Steve's website, but the results are coming from the IGG database.
Each row of the results contains two critical numbers: The film number in the LDS archives, and the record number on the film. So, for example, my great-great-grandfather's death certificate is listed as :
Greene Todres 60 y Dec 23 1901 37861 (1901) Manhattan 1840 - 1841 1323000
Next, I go to FamilySearch.org, which is run by the Church of Jesus Christ of Latter Day Saints (LDS). The LDS church has a deep interest in genealogy, for reasons I won't go into here, and they have an incredible wealth of records on microfilm. You'll need to register for a free account on their site, and then search for "PhotoDuplication Service". You are permitted to submit up to five document requests per month; I keep a prioritized list of requests that I plan to make and every month I send off the next five on my list.
So when you fill out the PhotoDuplication request form, you'll take the film number from the last column of the Steve Morse/IGG data -- but be nice to them and add in commas. So Todres's certificate is on film 1,323,000. And at the end of the form, they ask for the document number within the roll, so this one is 37,861 (1901). The other fields are self-explanatory.
Once you submit the form, it goes into a work queue at the main LDS Family History Library in Salt Lake City. Volunteers there will retrieve the given film, scroll to the given frame, make a scan of the document, and email it to you.
I have made some amazing discoveries this way.
They don't have the resources to do a search through the film, which is why you need to provide the frame number. But for older NYC records, this system works amazingly well.