Uh, no shit? That's how light works once you're able to travel at relativistic speeds - communication over interstellar distances using light is going to take ages.
Even within our own solar system interplanetary travel will have significant communication time delays.
Edit: also, we already know that matter and light can't exceed c, but I wouldn't be surprised if we discover that other forces (gravitation, or another that we haven't understood yet) can transmit information at speeds >c. I wouldn't be surprised if we turned to quantum entanglement for instantaneous communication over extreme distances either.
It might become like the days of sail. The fastest mode of communication might actually be the speed of ships. In order to get a message between earth and alpha centauri you might have to actually build messenger ships.
You might have to build small automated FTL capable ships with massive data storage capacity and then download all of the data you need to send and then set the ship off on its way.
Most people have missed the bit about time dilation messing up the clocks used in signalling, which I thought was interesting at first. However, surely the fix is just as simple as including a timing signal with the transmissions?
Jerry Pournelle's "CoDominium" books work like this. The ships are FTL, but can only use the FTL drive at a certain point to leave a system. There isn't a way to send messages faster than light, other than a ship. There is mention of "message sloops" which are small ships with high acceleration wich can move from the jump point to the inner system faster than one of the battleships.