As a lifelong hunter I thought I would post my thoughts…
The Dick Cheney hunting accident was while quail hunting, completely different than most deer hunting. With quail hunting you walk in a line while dogs work out in front of the hunters to locate and flush quail. Often times the quail flush and then fly at eye level, often to one side or the other. As you take a bead on the quail and are following it with your shotgun it is easy to loose track of your surroundings and swing your shot into the direction of other hunters to your side that have gotten out in front of the line a little. That is my understanding of what happened with Cheney. That is just an inherent risk of quail hunting and not something hikers need to worry about. Also most, if not all, quail hunting takes place on private land so even less of a consideration with hiking.
Most deer hunters sit and wait for deer to walk by either in a stand or on the ground. I guess there are deer drives too. The thing with people getting shot while deer hunting is it is usually a very young or very old hunter that shoots at brush moving, the sound of something walking towards them, etc. However that is probably .0000001% of hunters that would even THINK about shooting at a sound or movement. I can confidently say that there is no chance, ever, not even a slight possibility that I would shoot a human. I would never shoot something in the woods or even aim at something unless I was 10,000% sure it was a deer and I knew if it was a doe, buck, what size buck, etc. I wouldn't even shoot at a body of a deer without seeing what was on it's head much less just at noise. That is how 99.99999999% of hunters operate, they are not out there just waiting to unload on the snap of a twig. As in any activity, there is always going to be that tiny percentage of dangerous people but I am sure statistically you are many many times more likely to be killed by someone texting on their phone while driving on the way to the trail than to be shot by a hunter.
I know a lot of anti-hunters like to portray hunters as fat, drunk, Elmer Fuds stumbling into the woods after a night of drinking with a fully automatic assault rifle and a 12 pack of Budweiser just waiting to hear a snap of a twig and start firing wildly in all directions. That is not the case. In my experience most hunters are a little redneck, yes but they have also mostly been shooting and in the woods since they were 8 or 10, are careful about gun safety, sober, respectful, and would NEVER shoot at a sound or movement, not in a million years. I would venture to say though, that the hunters on public land tend to be the wild cards. With a lot of public land all you need is a gun and a cheap instate license so you do tend to get some less experienced people. I think the people that shoot people while deer hunting though are really the wackos or either too young or too old to be hunting alone. By young I mean shouldn't be handling a gun by themselves. By old I mean senile/dimenshia, legally blind/deaf older people that also shouldn't be handling a gun without supervision.
I think wearing orange or at least not brown can't hurt but at the same time, I think most deer hunting accidents where someone shoots a hunter thinking they were a deer is because they shoot at noise. Maybe wear some orange and talk or whistle as you walk or something if you are super worried about it. I think its just better to not worry about it and take my chances like I do everyday when I crank up my car to drive to work.