None of the quartet was selected for the last games of the season, and their contracts expire on June 30.
Hart and Legzdins become free agents – with fans of B read who wants to find the old man of Man City after an excellent season on loan with the club in 2009/2010.
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Legzdins has had two spells at the Blues, as a trainee, to come back under Gary Rowett, but for a second time in 2017.
The two were 33 years old and looking for their next club.
“They go to their new situation, whatever it may be, with our blessing,” said Dyche.
“All the players that I had during my stay at Burnley, I’ve always wished good luck and been there for everything they need if I can help.
“There is nothing, other than respect for the players who are gone. They have been brilliant servants for me, for themselves and for the team.
“Jeff has been a fantastic professional for us. He has held many different positions with a great attitude towards what it is like to be at Burnley Football Club.
“Joe is a professional fantastic. He lives for his football and his family and I wish him good luck and I have nothing other than positive about it.
“Adam Legzdins has not much featured but it is a pro fantastic and Azza – I’ve discussed it at length with him, and it was hard for me. I have a good connection with Azza and we shared some thoughts.
“There is no angle with these situations. It’s just a time where the decisions are taken. ”