Dum Dum Rounds, or Dum Dums, are power ups to the pistol which enhance the power to about ten times, making the pistol very deadly. It can take down enemies very quickly from the Assault Trooper to even the Battlelord. Because of this, the maximum amount of Dum Dums you can carry is only 36 rounds, which is 3/16 of the maximum amount of Pistol ammo in Duke Nukem 64, so it should be used only for areas with powerful enemies like areas with Assault Commanders and/or Mini Battlelords.
The Dum Dum Rounds are very rare, so conserving ammo for Dum Dum Rounds is a high priority.
These rounds are first found in Gun Crazy (or Red Light District) in a secret area next to the room with three buttons you must press them in the correct combination. Next to the three buttons there will be a poster to your right that has the word "RiotMaster" on it. Use the poster to open up a secret enclosed area with a clip of Dum Dums. Since you are captured by the Pig Cops at the end, you do not have to conserve ammo here, unless you are taking the secret exit to Duke Burger.
The second Dum Dum Rounds can be found in Toxic Dump in a secret area near the end of the level, however, to access this secret, you will need explosives to get in. After you've reached the room with acid current, ride the current 3/4 to the end, then try to find a crack on the wall on your right. Blow it as soon as you see it and enter the area with machinery. Look behind the pillar and you will see some Dum Dum Rounds.
The third set to find can be found on Launch Facility, and this is the first set that is not in a secret area. These rounds are also near the end. They can be found near the switch that opens the door to the acid current behind a round pillar. After destroying the space ship, go to the switch that operates the door to the acid. Face the switch and turn 45 degrees to the right and go towards the pillar. There will be some rounds for you to collect.
The fourth, and easiest set to find is on Battlelord, and is found right before you start. Before you start, quickly strafe to the left and you will find some rounds sitting on a platform. If you are maxed out on Dum Dums, you can kill the first boss in the level very quickly. The Battlelord will die after about 33 shots from the Dum Dums.