Emily Carroll is one of the singular talents in comics today. Her storytelling voice and artistic style are so unique and commanding that they make for a fully realized reading experienced, and her story Beneath the Dead Oak Tree is available for free on Gumroad! Here's the premise with the link following:
"Murder, decadence, cowardice, guilt, and aristocratic foxes in wigs all combine in this gorgeously poignant poem/folksong from Emily Carroll, about the futility and heartbreak one can run into when dealing with vengeance."
Beneath the Dead Oak Tree by Emily Carroll
This is a great introduction for anyone interested in unique comic book storytelling. Many of her works are available online for free as webcomics, and she has an amazing collection of stories called Through the Woods that I wrote about in a recommendation below:
Through the Woods by Emily Carroll (Margaret E. Mcelderry Books) - Emily Carroll may just be the best horror creator in comic books currently. "Through the Snow" is a collection of her selected works. Carroll's ability to infuse within her comics a palpable dread borne of primordial fear of the woods, the night, and the unknown is second to none. Not to mention, she makes amazing use of the webcomic medium and many of her works are available online for free! I would start with "His Face All Red" http://www.emcarroll.com. The collection is available on Amazon. Through the Woods preview pages on Amazon
You're very welcome! If you enjoy Emily Carroll's work, she put out a book a while back. It has His Face All Red and a few other stories in it.
Everyone here has already mentioned some good mainstream horror comics, so I'm gonna recommend a couple of short-story anthologies.
https://www.amazon.com/Through-Woods-Emily-Carroll/dp/1442465964
It's actually custom made! You can get it here if you want. It's 1366x768. It's from Emily Carrol's horror/fantasy anthology graphic novel Through the Woods, which is pretty good. I'd recommend it. Also check out The Prince and The Sea on her site. It's my favorite of her work.
And for all that, now you have to make assumptions about me based on my desktop!
You asked for non-violent thrillers, which is a pretty tall order, but too hard to come by. Emily Carroll's Through the Woods immediately came to mind. She does a great job writing suspense/horror without having violence or gore.
If you're just looking for non-thriller graphic novels without violence, any trouble finding them is probably genre related. Violence in autobio comics, for example, is very rare. Alison Bechdel, Seth, and [Jeffery Brown](http://en.wikipedia.org/wiki/Jeffrey_Brown_(cartoonist\)) are all great autobio writer/artists that have been active for decades now. Each of them has non-autobio non-violent works as well.
Last year's "best comics of" lists wre fully of great non-violent titles. A few off the top of my head (and the spines on my shelf): Mimi Pond's Over Easy, the Tamaki's This One Summer, Fumi Obota's Just So Happens, Bryan Lee O’Malley's much-anticipated Seconds, Stephen Collins' The Giant Beard That Was Evil. Through the Woods was last year too. It was a good year for well written, non-violent books.
If you're having a hard time finding good non-violent graphic novels, it might just be that you're seeking out publishers/writers/genres that are full of violence. There are more great non-violent books available now than ever before.