That IS a very interesting question! For starters, the oceans and the atmosphere would still be mostly spherical because of gravity. As a result, the continents would not be at all similar to the current land masses. The part of the earth not covered by water would probably look like 8 separate mountains (or pyramids) rising from the ocean, one at the each corner of the cube.

Sunrise, sunset, and length of the day would be very dependent on location, as large shadows would be cast by the cube faces. Depending on the tilt of the earth's axis, some of the cube faces may not experience sunlight for most of the year. This would have huge effects on climate and ocean currents, which would in turn determine which parts of the cube would be inhabitable by humans.