I echo Morritt's.
You can usually get a two bedroom, three full bath timeshare with a full kitchen for the week for quite a bit less than what a standard hotel room goes for. They are in the process of refurbishing the pool side units and most, if not all, should be done by June. The oceanfront units were all refurbished a couple of years back courtesy of Hurricane Ivan.
Here's a link to a forum with owners' posting of units for sale or rent. They don't allow posting of prices, but might be worth a look.
Morritts Tortuga and Grand Resorts Forum :: View Forum - Rentals and Resales
The Reef resort next door is quite nice, but more expensive to rent.
The biggest down side (or up side depending on what you like) is that these resorts are isolated out on the East End. You would be hard pressed to find someplace that's a longer drive from George Town without getting you car quite wet in the process.
Since we "own" there (I guess that's by way of disclosure), we get a discount from Tortuga Divers, the on-resort dive op, and always use them for boat dives. They have always been very good to my wife, her kids and I and have never given us any reason to try anyone else. I have heard nothing but good things about Ocean Frontiers, the other East End dive op, both here on ScubaBoard and from people I've talked to in person.
I assume that you've been checking hotel prices, so you're over the initial sticker shock. Be aware that everything is more expensive in the Cayman Islands than in the U.S. and eating in for even one meal a day can add up in a hurry.