It's when you're moving quickly and quietly along the trail that you're most likely to encounter a bear in the Rockies, and that's because they're easier to sneak up on that way. I run into more bears when I'm on my mountain bike than when I'm hiking.
Large predators use trails as often as people do. If you're not making enough noises to identify yourself as human, then you might not give the wildlife enough notice to get off of the trail ahead of you. While trail running, you're typically concentrating on breathing and don't talk a lot, you're also focused on the trail, so you yourself are often unaware of what's on the trail not far ahead.
Most of the time the bears will startle and run off the trail, but if you're cruising along, and round a corner right up behind a grizzly bear, it could act defensively. This is the scenario where you would play dead, the bear might paw at you for a bit, but if you play dead it'll figure it's taught you your lesson and then wander off.
Black bears are completely different. If you play dead with them there is a chance they're figure they just scored a free meal and start eating you. With black bears you either have to scare them off, or fight them off. Both scenarios are rare.
Best thing to do is to sing while you run alone or shout down the trail occasionally. But it's always advisable to run with a partner, someone you can talk to while you run, and who can help you tag-team intimidate a bear if you encounter one.
They sell small cans of bear spray at the till in the markets in the mountain towns. It might be wise to pick one up and run with a water pack to carry it in just to be extra safe, but I can pretty much assure you that 99.9% of the time the bear will wander off to a safe distance. That safe distance is quite a bit shorter in national parks however, bears are quite more accustomed to people there.
For the most part you don't need to worry about bears. You may see some, thousands of people do in the Rockies, especially in national parks. Enjoy your runs, just make sure you know what to do in the unlikely event of a dangerous encounter.