The short answer is no, you do not need to be overly concerned about watering with city water.  Go ahead and do it.  But you are correct that the chlorine in the city water will kill some of the microbes.  That is what the chlorine is there for in the first place, after all, and unfortunately.
In a perfect world, we wouldn’t want the chlorine, as chlorine is clearly not part of the natural process.  But in a garden setting the soil is porous enough, and the microbes should be plenty enough, that the bulk of them wont we affected.