Think about it for a moment. How can one speak one’s peace? Peace is quiet, isn’t it? If you’re speaking, you’re not quiet. And what you have to say might well disturb the peace. I can hold my peace at a wedding (“speak now, or forever hold one’s peace”) to maintain the decorum and not embarrass the bride or groom (or their parents, or their aunt Maisie, or the dog . . .).
I can speak my piece, though. Perhaps it’s a piece I’ve memorized, or perhaps it’s a piece that just comes to mind during a heated conversation. Usually when we say someone spoke their piece, they weren’t necessarily being kind. “I went to the board meeting last night and spoke my piece about their stupid plans.” A common variant of this is “say one’s piece.” The meaning’s the same. You speak. It’s anything but peaceful.