A Rule? Nah

If you are the server in tennis, you have control of the flow of the match. With that control comes responsibility.  

Prior to beginning your game, announce the score of the match loud enough that all involved are alerted and may, at that moment, challenge the score if they believe it to be incorrect. 

After the score announcement, if it goes unchallenged, make eye contact with the intended receiver to be sure they are ready to play, and only then begin your serving process. 

After the point ends, call the score loud and clear, then immediately collect any balls that are near you and proceed to position yourself on the correct side(deuce or add) to serve the next point. 

If there are balls on the opponent's side of the court, raise your hand and let them know you need either one or two balls to begin the point. After you have two balls in your possession, announce the score again, then once more make eye contact with the intended receiver to be sure they are ready before you trigger your serving process. 

All of that can and should be accomplished in 30 seconds or less. 

If you like to have (a personal preference) all  three balls in your possession before you serve, let your teammate and opponents know that before the match begins. 

If getting that stray third ball to you is not convenient (let's say it rolled 50 feet away to the side fence on the adjacent court), don't wait around for it to be retrieved and given to you. Place your personal preference (That’s all it is; it's not a rule that you can DEMAND the third ball) aside for the moment and proceed with your service game. When the natural flow of the game, or at the end of the game, that ball can be retrieved and not have interrupted the continuous play rule. 

On another note, if your serving ritual/process includes bouncing the ball before the toss and swing, a couple of bounces will suffice. Please don't be that player that puts everyone in a coma waiting for your lucky number of bounces to be performed before starting the point. 

Keep it moving. Keep your opponent informed (say the score). And KEEP IT FUN!