By knowing the angle it enters the cube. If you know what angle it came from you know which face it had to pass through to get there, trigonometry will come in here
Do you have an example of how I could use trig in this way?
By knowing the angle it enters the cube. If you know what angle it came from you know which face it had to pass through to get there, trigonometry will come in here
Without knowing many relevant things about your engine I'd say picking is pretty ace instead of messy or limiting. You can find all your objects, voxels and gui elements with checking a single value, and your user is guaranteed to pick exactly the pixel he chose. If you do this by rendering to an offline ID framebuffer, you can even do it with practically no performance hit.
The math isn't overwhelmingly hard either, so you can just raycast through the voxels. And it doesn't involve reading back from the GPU, which is a nice feature. Lode's tutorial has a very nice explanation which you should be able to turn into 3D pretty easily.
You can find the surface the ray huts by taking the position of the hit (i assume you dont use some laggy raycast which steps in tiny steps and each time checks what voxel its in...) and checling its distance from each 3 planes pointing along an axis.
Like if the distance of the point from the xz plane is about 0.5 (half voxels), it will be on one of the y axis surfaces depending of the sign (either -y or y surface)