Cube-Mesh Collision Detection

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I'm making a program that converts a 3D model into a voxel model. Most of the programming is done, but when I test an area (cube) for polygon mesh, at the moment, it only test for the models vertices by testing whether the point is in the cube.

To properly convert the model I need to test whether the polygon mesh intersects with the cube. I don't have any real experience with collision detection but because this is a converter there isn't much need for speed so it can be worked out by brute force, which I assume would make things easier.

It's a simple question of collision detection, but how do I go about doing this?

Any help is much appreciated.
Thanks,
Rowan.

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