**Due to a comment by z 4assed monkey criticizing my choice of words in my original solution, this solution was edited on 7/4/2015**
This really isn't a difficult achievement. It is easiest to do the achievement on either Prologue, Chapter 1, Chapter 2, or chapters 1-3. Chapter 4 and chapter 6 both have areas where you are likely to take damage from smoke/fire, so the early chapters are the best place to complete three consecutively without taking damage.
Play on easy and basically just take a stealth approach. The enemies are far less observant than on the other difficulty levels. Save often and load your save if any soldier has a red eye over their head, because this indicates you are going to be attacked, and you will likely take damage. Also, (obviously) load your save if you take damage from anything else. It may be a good idea to check your health level before saving.