It cannot be done correctly with just a soldering iron and you risk damaging the circuit board. Rosin core solder is also verbotten. The rosin acts like a slow motion acid that will damage the board.
When done, I simply wash the board with readily available flux solvent spray. j
To do it correctly, the board is warmed slowly with controlled temperatures, rather than fast and hot. Typically, the underside of the board is warmed to just below the melting point of the solder, around 150C, then the component side is heated simultaneously and the components is removed, then there's a cool down period as well. All this is done thru high magnification microscope with a mechanical mechanism to place the component in the exact position without displacing surrounding components.
If I could upload a 4 MB file, I would show you a Powerpoint presentation one of our staff did that shows how to do it properly with temperatures, etc. Here's link to the machine we use with a video showing the machine in action: http://www.pdr-rework.com/node/69
One must realize that a compromise must be made when the need to prototype, repair or, in the case of the Radio Amateur, homebrew a single unit for one's own uses.
While I have used nothing much more than a good needlepoint regulated iron, headband magnifier, tweezers, eutectic solder, solder-wick, chipquik for certain removal tasks that involve more than a two pin device in prototyping various boards in development, initial testing or initial programming work, those prototypes are NOT what happens with final production boards at all, that is a "next stage" production issue and is typically jobbed out to outfits that specialize in SMT board population and manufacturing techniques such as you quote here. As well they should.
That said, I have plenty of examples of SMT prototypes, one-offs for my own uses, the occasional one-off custom work that price precludes use of mass production techniques, all have weathered the test of time very well, all were assemble by hand by yours truly just as described.
And try your best to make 'em all look pretty.
I have always maintained that the key issue in hand soldering of any type is resolved with practice, practice and more practice at the art and science of hand assembly soldering.