If I didn't ride on roads I wouldn't be able to ride much at all, really. That's all there is to ride on here, roads and occasionally the pasture you might happen across lacking a fence or with the gate open. Best avoided during hunting season though. I've only ever had one problem with a dog coming at my horse, some 8 years ago this old lady's lab decided to run up on my mare. It got a prompt kick in the face and has not ever bothered myself or any of my horses since. It's still around, old and grey and missing one eye. (Unrelated to the horse kick)
Not that there aren't dogs here, they're just used to horses and cattle and the horses are used to them as well. I always take my own dog or dogs with me as well. On the topic of cars, thankfully no one has tried to run me over or anything, though one idiot was speeding along and slammed on the brakes /right/ on my gelding's rear. We got showered with gravel and when he crept past at 2mph my gelding was having none of it. I think the other worst ordeal is when some minivan came up beside a greenbroke mare I was riding, pulled over and a lady promptly got out with her 4 shrieking children that swarmed us and wanted to 'pet the pony'.
I HAVE however been attacked by other horses while riding the roads though, crazy neighbor had her two horses loose in her front yard to graze. After the fact she yelled at me about how dare I ride down the road past her house and 'upset' her horses. (I ended up having to bail off my mare onto gravel to avoid getting a hoof to the head from her gelding and my mare was run off a mile down the road.) Thankfully no major injuries were suffered.