Are the Fable games' stories continuous?

I've played through Fable, and I own Fable 3, but I want to know if the story will still make sense if I skip Fable 2. Do all three have one continuous story, are they completely separate, or something in between? Are there things I won't understand in Fable 3 if I don't play Fable 2?

Please avoid spoilers.

The story of Fable is about the family of the hero from the first game. Each subsequent game is a continuation of that hero's family that comes into their own powers and finds their place in the world as they discover their abilities and stop some great evil that is currently threatening the world. While they all share the legacy of the family, you do not have to play the games in any specific order for lore purposes.

I played Fable II thoroughly, and later started Fable III when it came out. As far as I could tell there wasn't really enough related to Fable II to really matter. If you want my true opinion, play Fable II instead and forget about Fable III. Fable III was awful by comparison. Then again, I know that's not what you asked.

I don't think you'll have any real continuity problems if you skip Fable II. I never played the first Fable but greatly enjoyed Fable II, and I think I heard that the first two are more connected.