Can airplanes transmit/relay flight data to ATC's periodically while in flight? The flight data recorder is one important piece of information in the event of a crash. Can this information not be transmitted or relayed to the ATC's while the flight is air borne so that they don't have to wait to recover the data recorder after a crash ? I mean to say that periodically the data can be sent back to a ground controller when the signal strength is high and it can be logged. Please let me know if there are any limitations to the amount of data that an air borne flight can relay back to the controller.
No, there's no data relayed like this to ATC. All ATC really gets is the filed flight plan, and then the transponder data. ATC just cares about organizing all the traffic in the skies, and making sure it is safe, orderly, and expeditious.

Some airlines use a system similar to this to relay maintenance data to the airline from the aircraft, that's how AF447 was found to have something wrong, and how they generally found it's location quickly(I think within 48 hours debris was found from the a/c due to the location info through ACARS(Aircraft Communication and Reporting System)). This system, and it's capabilities are optional and variable depending on what the airline wants to spend. This is why there's so much controversy over the loss of MH370, since they didn't have the full reporting schedule, and a very basic version.