When looking to get a 19" monitor for free one should no longer be looking at fixed sync monitors tossed out by Universities (as talked about in the above writeup by Deimios which should certainly be kept for the future since it does still contain valuable information about hooking up fixed sync monitors). There is nothing inherently wrong with those monitors, it is simply that they went out of style such a long time ago that they aren't really being tossed out in large numbers anymore, and thus you will largely be wasting your time if you are hunting down auctions and trash bins looking for them. Even if you do find one you might very well spend more on the cable than the monitor is worth.
However (as of 2008) there is one very easy way to procure yourself a free computer monitor without having to deal with any rare fixed sync monitors or wonky cables. Just start asking everyone you know if they have an extra computer monitor you could have. Monitors are worthless now that computers have been around long enough and enough people have been dumb enough to replace perfect monitors (and printers too) when replacing their computers. There are literally millions of monitors sitting unused in closets and basements, particularly now that LCD monitors have come into fashion, many people have even relegated those formerly high end 19" and 21" monitors to the basement.
If you absolutely cannot find someone to give you a free 19" monitor then go on your ebay and search for local ones. Absolutely perfect ones that can be picked up will end without bids for starting prices rivaling that of a candy bar.