I prefer to grab my opponent's piece, take it off the board then move mine to capture. I move reasonably fast, and try not to show my feelings, unless I'm happy. Then sometimes I'll smile. To offer a draw, I saw "draw", but if the situation is more formal I shrug my shoulders before I offer the handshake and I won't say anything. I may drop the king to resign, but just shake hands when I resign formally.
How do you move pieces in physical over the board chess?
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Move confidently when you aren't.
Move without confidence when you are confident.
Move angrily when you are happy.
Move happily when you are angry.
Move up when your game is going down.
Move down when your game is going up.
Move like Jagger when you are Christina Aguilera
Move like Christina Aguilera when you are maroon5
When you are not Bobby Fisher, move like Bobby Fisher.
When you are Bobby Fisher ... damn, I'll stop giving you advice.
I don't mean what moves you make, I mean the manner you move and body-language.
I notice new players tend to move the pieces meekly as if they aren't confident. And when new players capture they tend to move their piece onto the square of the captured piece, slowly moving the captured piece to the side with it then removing the captured piece.
But more experienced players tend to move a lot more confidently and faster. And when capturing a piece they either first grab the piece they are going to capture before moving their own piece onto that square, or do this thing where they move their own piece onto the square of the captured piece and kindof snatch the captured piece off the board all in one movement with a rather satisfying sound.