I feel this is legal. I'm using FIDE rules.
This is the rule that describes what happens with a capture:
3.1.1 If a piece moves to a square occupied by an opponent’s piece the latter is captured and removed from the chessboard as part of the same move.
There's nothing there about what happens to the captured pieces, other than they are removed from the board.
Now, of course somebody will argue that this rule exists:
6.2.3 A player must press his clock with the same hand with which he made his move. It is forbidden for a player to keep his finger on the clock or to ‘hover’ over it.
And there it says "with the same hand". But you also have to remove the captured piece with that hand, and once you've done that you have to press the clock, and there is no requirement to let go of the piece. I argue that this rule is intended to be used against people who move a piece with one hand and press the clock with the other, not against using the piece that you happen to have in your hand.
Of course, there is an opportunity for bad behaviour, by hiding pieces in your hand so that the opponent doesn't notice he is behind in material (IMO that's that opponent's fault, look at the board not at the removed pieces), or worse, hiding the piece that the opponent wants to promote to. Or by simple distracting the opponent by constant fiddling with pieces in your hands.
But there are already rules against distracting the opponent, there is also the umbrella rule against "bringing the game of chess into disrepute", and when you want to promote but don't immediately spot the piece it is always acceptable to stop the clock, call an arbiter and have him fetch the piece for you. That is enough ammo to use against bad behaviour.