One of the simplest line configurations in space just utilizes the parallels to the coordinate axes that pass through the (red) points with integer coordinates.

If we want to avoid the triple collisions at all these points, we can shift the lines one half unit each, like so:

This results in a dense packing of cylinders. Another possibility to avoid the collision is to let the lines spiral around the red points. I haven’t found a nice way to do this because the three helices would need to pass through the eight cubes surrounding a red point, meaning this is impossible in a symmetric way.

However, there is another line configuration where the lines pass through all the main diagonals. This is more complicated, because we have now four sets of parallel lines. Again we can shift the lines to avoid collisions.

Now, with four lines through each intersection, we can replace them by helices in a pretty symmetric fashion.