By no means do all BYCers have a 'top' on the run (neither solid nor wire/netting)... although a lot do, because there are a whole lot of pet or hobby oriented people here who really REALLY do not want to lose birds if at all avoidable. Whereas chances are good your grandfather's attitude was more "well, as long as most of them survive most of the time...".
Without a top on your run, birds can leave if they want to (will they want to? very hard to predict, but with a non-huge run that goes to dirt and just a 3' fence I think most chickens would be at least sorely tempted) and hawks can eat yer birds whenever they want to.
It is really pretty easy (in non-snowy climates or the non-snowy part of the year, anyhow) to stretch deer netting or bird netting over the run, and that will pretty much prevent hens flying out or hawks getting in (much).
But, the bigger problem with a 3' fence is that in most cases it will leave your birds very vulnerable to dogs, which are arguably the worst daytime predator (because a hawk will just get one bird at a time, whereas a loose dog or two can kill your entire flock in minutes). THerefore unless this coop will be in a totally COMPLETELY dogproof-fenced backyard and there is ZERO chance the gate might ever get left open at the wrong time, I would really suggest a more serious fence. Although honestly even a strong 4' fence can be climbed or jumped by some dogs; higher is better, depending on what you percieve as the likelihood of loose/stray dogs around.
Good luck, have fun,
Pat