You can use a sprite sheet of the character walking to their destination from a top-down perspective until they get to water (I'm assuming these islands have water around them). When they get to the water, you can use a sprite sheet of them swimming freestyle or backstroke, etc, again from the top-down perspective. If the islands are about the size of the player, just have him dive in and swim, no walking required.
As an alternative to swimming, you can have him picked up by a submarine that just so happens to surface at the right place and time and have that carry the character to the next island while still on the surface. Sub reaches the island, the character disembarks and the submarine submerges again.
I do like the idea of something like a jetpack, or flying carpet as well. Hell, you can even mix it up a bit: submarine this turn, jetpack the next, followed by a giant slingshot, etc.
However you wish to do it, I reckon sprite sheets would be the way to go. Just be prepared to do a lot of editing (unless you're crazy talented).