Jake, Ryo and Charlie from Moorfield Junior School have explained how to cut up a 3x3x3 cube using 6 cuts:

We got a 3 by 3 cube and then we cut it 2 times to make 3 lots of 9 cubes.
Then we piled all the cubes on top of each other. Then we took another 2 cuts to leave 9 towers of 3 cubes.
Next we layed them next to each other. After that we took another 2 cuts to leave the 27 unit cubes.

Chris B, Elliot and Joseph, also from Moorfield Juniors, sent us a diagram to show where these cuts should be:

Do you think it is possible to do it in less than 6 cuts?
How about the 4x4x4 cube? The results are rather surprising - do email us if you crack it.