Air Traffic Controller reported releasing an IFR aircraft that did not fly the assigned route resulting in flight below the Minimum IFR Altitude for that area.

Narrative

Flight Data called requesting a; IFR clearance for Aircraft X from ZZZ to ZZZ3. I relayed the clearance to Flight Data with a revised routing VIA ZZZ1.ZZZ [Airway].ZZZ2..ZZZ3. Flight Data correctly read back this clearance to me; the aircraft was advised released for departure.On departure Aircraft X checked in out of ~2000MSL 'runway heading'. I found this unusual but assumed it was because they had just lifted off and may not have been in controlled airspace yet. A few minutes later Aircraft X requested to fly direct ZZZ1. This confused me further because proceeding direct ZZZ1 was part of the departure clearance. I told the aircraft that since it was part of their departure clearance; they could make that turn at their discretion. I did not explicitly clear the aircraft direct ZZZ1 due to still being below the MIA (Minimum IFR Altitude). After the Aircraft was established direct ZZZ1 and above terrain; I inquired why they were initially flying runway heading on departure instead of turning direct ZZZ1. The pilot responded that they were cleared to ZZZ1 via radar vectors in their departure clearance (I did not specify radar vectors in the departure clearance). At that point; after informing management of the discrepancy; I handed off the aircraft to ZZZ approach and transferred communications.Management later informed me that my departure clearance as well as my hearback/readback of the one repeated to Flight Data was correct. The pilot of Aircraft X incorrectly read back 'radar vectors' to Flight Data and this error was not caught.The root cause was a hearback-readback error during a conversation between Flight Data and the pilot. I have no recommendations to address the root cause.Addressing a similar situation in the future; I could be quicker to recognize an unusual situation on departure and issue corrective instructions more efficiently.