Generally, it is difficult to remove ribbon cables from drives and avoid the problems flamingweasel describes. There is a tool that can solve this problem, as I learned one night while rummaging through my tool box, looking for the needlenose pliers. The solution is canvas pliers.
Canvas pliers, also called upholstery pliers, as well as a few other names, are designed for stretching canvas (for oil painting) and for upholstery work. They are available at most decent art supply stores for about US $20. Canvas pliers have a jaw that is about 3 in. wide, and deep enough that one does not crimp the cable.
The canvas pliers are used to grab the entire width of the plastic connector, so that stress is distributed evenly over the whole connector, instead of just in one spot. The pliers are also better because, when used properly, they are pulling on the connector, not the cable, and the connector is designed to take more stress than the cable. You can coat the jaw of the canvas pliers in some sort of rubber or plastic to better spread the stress over the whole plastic, to get a better grip, and to avoid scratching up the plastic of the connectors, if you are finicky about this sort of thing.
A fair, though not great, substitute is two pairs of pliers, one at each end of the cable. This is generally more difficult, especially given the cramped conditions inside most computer cases, and the need to coordinate the actions of both hands.
Disclaimer: Canvas pliers are almost always iron or steel. These conduct electricity. Don't be stupid.