From GoToCayman.com - the Wiki for Cayman
Currently, the main telecommunication provider is Cable & Wireless. Visitors can access the Internet by dialing 976-4638 or 266-6328 and connecting with username "easy" and password "access". The rate is CI$0.12 a minute which will be charged to your phone bill. See http://www.cw.ky/internet/products/visitor-services/easy-access/ for details
There are a number of Internet Café's on the Island; Shooters Billiards Lounge and Internet Cafe, Café del Sol by the Cinema in West Bay Road, PD's Pub in Galleria Plaza, Dickens Coffee shop in Galleria Plaza, Chelsea's Sports Bar in Queens Court, Big Daddy's on West Bay Road, the West Bay Polo Club, PC Powerhouse in the West Shore Centre on Walkers Road, Ye Olde English Bakery in the Dockside Building in Mary Street, and Sunset House. If you staying in the East End area, then The Thirsty Surfer located in the Reef Resort has an internet Café. Cable & Wireless also have an office in George Town across from the harbour where you can access the internet. However, the cheapest internet access I have found is two terminals in the "Coffee and Bite" store just behind the Post Office in George Town; CI$6/hour.
If your ISP is a member of the iPass scheme, then you should be able to use Cable and Wireless; see the details at http://www.cw.ky/internet/products/broadband-services/internet-roaming/.
For those looking for a wireless network, then the Grand Cayman Beach Suites (ex Hyatt) was the first hotel to offer a Wi-Fi network that covered all of the resort; guest rooms, poolside, on the beach and on the golf course, as well as meeting rooms. Charges are US$6 per hour or US$17.99 per day. Their network will be expanded to cover Rum Point at some stage. Guests need to register with the front desk to obtain a username and password. I have also received a report that the Marriott have an unsecured wireless network access point in their lobby that is therefore open to all!