I ignored this thread the first time I saw it for pretty much the reasons alsmost everybody mentioned above, i. e. 'so what? SAren't thrillers supposed to be all about the threat of violence?'. Since then, I've mapped out the plot to a heist story I've been contemplating for a while, then decided it needs moresuspense, so now I'm thinking along the same lines because I want it to be as non-violent as I can get away with. The story started as a con story, and the cool thing about con stories is that the protagonists don't outgun their antagonists - they outwit them. It's why I want to write one.
So now I came across the thread again, and this time I'm listening. Well, reading. You know.
So, a possible answer to the OP may run along one of these dimensions:
- The threat of violence vs. actual violence: Nothing wrong with the MC fearing for her life. Pegleg Pete tried to kill Mickey Mouse over and over, didn't make the stories any darker. They do change their flavor once you actually show the violence. Knowing there's a killer on the loose is one thing, seeing him dismantle his victims is another. I can do without the latter, keep the former and still call it a nonviolent thriller. In 'The Man Who Knew too Much', the threat is that the villains kill the couple's son if they don't assist in that other murder, and the suspense comes from them traing to avert both murders. Good enough for me.
- Strong or explicit violence vs. the less outrageous type: So the bad guy has a gun, or a knife, or even an ice pick. Generations of directors got away with a minimum of bloodshed, although recently that seems to be out of fashion. I can do with a lot less gore. And it's not just a matter of taste: the more the violence becomes the normality of your story's world, the less of an impact any violent act will have, and the less suspense you'll get from it. (Personally, I've groewn to find movie gun fights extremely boring. Unless you're John Woo or someone like that, I'd rather watch a good brawl. But that's beside the point.)
- Violence against the MC vs. violent MCs. It's a different game. You live by violent rules, you accept that violence is what's coming to you. If the MC reacts violently to a violent threat, the violence escalates. While that can be fun to watch, it isn't necessary for the suspense. Unless the threat comes from something very different (dunno, maybe deafness from all the shooting). If the violence is one-sided and doesn't escalate, well, that may be one way to write a thriller that's not about the violence.
- the nature of the threat: Medical thrillers and natural diasters have been mentioned before. There can be a threat of insanity, of being deported, of an evil conspiracy taking over the world, of hackers rendering you legally dead and taking over all your possessions and just recently, we've all become very aware of how real and all-encompassing surveillance is. There's a whole sub-genre there - the paranoia thriller. Not to mention countless supernatural threats.
Yeah, I saved that for last because otherwise there wouldn't have been much point to mapping out the other ones.