Air carrier Captain reported GRI Tower informed the flight crew of pushing into the movement area without clearance and that this is a recurring issue.

Narrative

After an engine start at the gate we pushed back for a crossbleed start and taxi out. We completed the tasks and received taxi instructions from Tower (they were handling everything on Tower frequency). When we started to taxi; I believe it was a Tower Manager or Supervisor told us that we had pushed into the movement area. We apologized. I mentioned we used to power out and this was the first time I had pushed out. He stated that he was aware that we had once powered out and said something to the effect that this has happened a lot lately. Also that the people that were pushing us were instructed to inform us to call for push; which we didn't get. Despite that; ultimately it is my fault for allowing the aircraft to push into the movement area. Tower informed us no harm; no foul" which was kind; but there was a foul. My fault. Biggest reason to write the report is so perhaps pilots can be given a bulletin to watch out for this issue; only because the Tower stated it has been happening lately (pushes into the movement area). I incorrectly thought there was space to push but was incorrect and it appears others have had the same issue."