On Grand Cayman, a tour to Stingray City is a treat. You have the opportunity to stand in shallow water to view, feed, and hold these friendly creatures. Moby Dick Tours is very good.
A half-day tour in the morning leaves you free in the PM. Good duty free shopping in town.
On Cozumel, there are dive and snorkeling tours. Otherwise, the island is mostly unattractive, not very interesting. If you're not into diving/snorkeling, enjoy the ship.

We love going to the beach and usually get a taxi when the ship docks and go to a beach. You can also get a taxi driver to take you for a "tour" of the islands.
Cayman has gorgeous beaches. Don't be talked into going to the turtle farm and town of "Hell".....unless of course you love turtles.

"Otherwise, the island is mostly unattractive, not very interesting. If you're not into diving/snorkeling, enjoy the ship."

That is really a very sad thing to say.

We love Cozumel and have been 4 times. Take a taxi from the pier and go to Chankanaab for a beach day and snorkeling. There is a resturant and bar that provides chair service. There are several other beach clubs also, but we found Chankanaab to be the least crowded.

Also look at activites offered by Aquaworld:

http://www.aquaworld.com.mx/cozumel.html

There is a bar hopping tour that is offered to see the "wild" side of the Island.

What dock are you stopping in Jamaica? Ocho Rios or Falmouth? Rafting the Martha Brae, tours of the Coyba and Shaw Park Gardens, the Gree Gotto Caves, Dunn's River Falls are good options. However, if you don't wish to use the ship's tour here, definitly hire a private driver. Not an area to just walk off the ship and try to do things on your own. IMPO.

Look at the ports of call boards at cruisecritic.com A wealth of information.

We’ve just returned from a cruise that stopped in Cayman. Like you, we don’t like the cruise trips, we prefer to do our own thing and get better value for money. We found this article on things to do in cayman as a cruise ship passenger really useful www.explorecayman.com/grand-cayman-shore-excursions/ If there are lots of cruise ships in the port the day you’re in then you will find that the main attractions will be very busy –however there are beaches and activities to be found that aren’t crowded, you just need to know where to go! It is a stunning island though!

My (non-traveling) sister did that same itinerary.

On Jamaica they stayed on the ship and were glad they did. While I personally like Jamaica, Montego Bay for cruiseship passengers just doesnt' make a very good impression. You really get hassled.

In Grand Cayman they did a ship-sponsored sting ray swim that they really enjoyed. In Cozumel I had researched briefly, printed off some maps and ideas, and they simply took a taxi into downtown, got some peso, went to a local museum, had a nice lunch, did a little shopping.

They were very pleased with how things turned out each stop.

we did this cruise...in 2008 and we had a great time...first stop was Cozumel...we went to chankanaab park and swan with the dolphins...next stop was grand cayman and we went to sting ray city...it was a great experience...next we went to montego bay there we just got of the ship and shopped duty free store got back on we where suppose to stop ocho rios but a hurricane changed the plans...our last stop was haiti we really enjoyed the cruise and we booked everything on our own before we left home...cozumel to start before the park we went to the little town to look around they bugged you so much to come into there stores that we got a taxi and went to the park....
I hope this helps you a little bit more....

In Grand Cayman stingray city is the place to be, You can do it on the Buccaneer Catamaran or another great option is the Outrage Eco Tour, which is a private boat. The wave runners tours to stingray city is also really cool.

In Cozumel, we did the Cozumel Beach Break with Island Marketing and it was amazing.

Where are you going in Jamaica? Which port? Montego bay, Falmouth or Ocho Rios?