OnTriggerEnter - Objects pass through each other?

I have 6 primitive cubes stretched out to form the "walls" of a large box. Inside the box, I spawn some number of primitive spheres with random velocities. I have colliders on the cubes ("walls") and on the spheres. I have rigidbodies on the spheres but not the cubes. When I press PLAY, everything works fine. The balls spawn and go bouncing happily about the inside of the box. But, if I set the tick mark for "Is Trigger" on the spheres, all my spheres go sailing right through the walls of my box. Nothing is being destroyed... except the physics of the collision itself. I'm not sure what is causing this or how to prevent it.