I'm a big fan of humane traps (it's how we got rid of 3 adorable mice who were living in our apt years ago) but if you have an actual barn, I don't think humane traps will do you much good, because you'll just get more mice. Any open spaces you have nearby will have plenty of mice who will come take their place.
I would prefer getting outdoor cats to getting poison. At least it's a natural setup. I'd agree with Alex - don't get too attached to the cats, and if another animal decides the cat is dinner, it's part of the circle of life for a rural setting. If you'd be horribly upset by a cat that disappears (either by being eaten or running away) then this is not the option for you.
Are there other natural mice killing animals you can look into? A pet owl or hawk would be awesome.
