Poyet left his post in June after being suspended by the club the previous month.
Buckley has stated that the players must now look ahead under Poyet's successor Oscar Garcia.
"I owe him [Poyet] massively to my career and a few of the other players do to," Buckley told Sky Sports News.
"But that's football these days. New managers come in seemingly every season, and hopefully Oscar will get a good crack of the whip. Obviously you get close to a manager. All the lads were. It's a sad thing that he went.
"But we can't let it affect us. We've got to look forward and move on."
Poyet guided the Seagulls into the Championship playoffs last season, where they lost to Crystal Palace.