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Have you seen upside-down?

Trailer:

Pretty interesting movie.  They solve the gravity problem — twice.  Of course, by “solve” I mean in sci-fi plot terms.  I guess from a writing perspective it’s a lot like a magic system in fantasy – as long as you establish clear rules for what it can do and its limitations, you can get yourself into all kinds of interesting situations.

@cathy back to your original idea, you could also go the steampunk route as well – maybe zeppelins or propellers sticking up everywhere on your islands.

Or maybe there’s some rare mineral or something that makes things float?  That would be helpful in causing geo-political tensions to add plot conflict.