Echoing Nate and rbasin, the basic problem was geography and logistics. The way Grant eventually did it was one of the riskier approaches; his preference may be seen in the first attempt at Vicksburg, where a waterborne flanking movement by Sherman was to be coordinated with a primary overland advance south from Memphis. However, Earl Van Dorn scotched that idea with his destruction of Holly Springs (and, to at least an extent, Forrest's operations farther away had an effect as well, although they're not usually directly considered with the Vicksburg operations).
Grant's army would have been at the end of a very torturous and easily-disrupted supply line (he was not literally operating without a supply line, despite his statements in his Memoirs, although it in truth was a very abbreviated and minimal supply line), and in fact did come close to exhausting his supplies until the Walnut Hills ("Chickasaw Bluffs") came under Union control and supplies started flowing via that route.
Porter was also very reluctant to risk 'stranding' a significant portion of his fleet below Vicksburg unless the operation looked like it had a good chance of success (and even then seems to have required some prodding from high levels to go along with it).
Sherman was never a fan of the operation as it was eventually carried out; he advised trying the overland route (presumably with better guards at Holly Springs) again, but Grant seems to have decided that to return to Memphis would look too much like a retreat, whether it was one in truth or not... and, frankly, might very well have led to Grant's removal from command (to judge from the near-contemporaneous examples of Burnside, Hooker, and Rear Admiral Du Pont)...