Tried using a Redstone Block to power the chain, no effect.
EDIT: After breaking the Redstone Block under the last Command Block and replacing it, it started to function properly. There's clearly an issue of the Command Block not being updated that it is powered.
I think it works like this: If the block is powered before input of a command, then it won't function. When the power is applied after there is a command then it works.
EDIT2: So I realized that the Chain Command Block doesn't output it's command unless it is powered with an impulse or a repeater command block. I think this bug is an issue with the repeater block + chain block + redstone block. I'll be doing more tests. Changed the bug title accordingly.
EDIT3: OKAY I GOT IT! Reproduce:
1) Place Redstone Blocks where the chain is going to be.
2) Place the Chain and Repeater/Impulse Block
3) Power the Repeater/Impulse Block
4) If there is a command in any of the chain blocks after the first one, they wont execute.
Changed Title again.
EDIT4: Video: http://youtu.be/j73JWzjmR34