You can use the Path -> Division functionality to cut your image into pieces.
Put the cutting grid on top, and make it a single object (Path -> combine).
Then select the cutting grid + the object below, and do Path -> Division.
If you have multiple objects that need to be cut, consider combining them first, or using the Multi-bool extension, else you'd need to duplicate the grid before you do the division - it will vanish after use.

Off topic:
Using 'peek' now to record gifs - works wonderfully on Ubuntu (http://ppa.launchpad.net/peek-developers/stable/ubuntu)