Greetings, playing a single player game here and I've run in to a problem. The AI has created 3 cities that are on 1x1 islands, completely surrounded by water. How would I capture these cities?

Fire

you'll have to attack form ships. Bring lots of siege weaponry.

Step 1) Bombard their defenses down to zero with ships.
Step 2) Have lots of units (espically seige) to sacrifice.

Optional step 3a) Try and get a few units promoted with the amphious. They won't suffer they attack penalty for attacking from a ship, Makes taking an island city far easier.
Optional step 3b) If it's late in the game, build lots of marines (navy seals if your american) because they start with amphious.

Yep. Bring your transport ships and some escorts. Use escorts to bombard down cultural defense. Then send in some siege weapons and attack. You will need a couple of units more than usual, but not too badly.

Also if you can make Level 3 units from scratch (10XP - not that hard) some amphibious attackers will help. You need to Promote Combat I and II, to unlock Amphi. Late game planes can help you safe some siege.

If you're already having a war with this AI and is still not ready, just blockade these islands. One to two ships per island is good enough to avoid too much war weariness. A blockaded single-tile island with no access to seafoods is rendered basically useless. In the mid-game you can use pirateers to do this job without engaging in a war.

What era are we talking about here, and what naval technologies do you have access to?

Thanks all, I had no idea you could use a ground unit to attack from a ship :)

Yes you can attack directly from the ship. There is a penalty if your unit does not have the amphibious bonus. Just do like the others said, bombard the city to 0, attack with your siege weapons then send in the mop up crew.