Problem: Long combat turns, and overly tactical battles
One of the problems with the initiative structure I have is that it makes a battle feel more like a chess match. Each person or monster rolls a d20 at the beginning of combat and then goes in order based on the rolls. With small parties fighting small groups of monsters turns can go quickly, but with a party of 5 fighting a legion of orcs, combat can get bogged down. We all wait for each player to take their structured regimented turn. This all has its place, and I know some people really appreciate a tactically minded battle. For me, I appreciate a battle that is fluid, messy, violent, and quick.
Solution: Team Initiative
In my experience one successful method to speed up a battle is to incorporate team initiative. In this version of initiative, the monsters and players (or their duly elected representatives) each roll dice to see which team goes first. When it is the players’ turn, the players elect the order that their team moves. After every player finishes their turn then the monsters get to go. The players are actually quite able to decide for themselves when to go. I think the players like it too; they are no longer waiting in a long initiative order to figure out what they want to do. They see the board and just act. After getting used to team initiative, my players (especially the fighters) were able to carry out their moves simultaneously. When they know the AC of the monster, they just report the damage or miss (along with glowing descriptive accounts of their sword or axe play).
The team initiative also encourages players to get into their character. It allows the character who plays the barbarian that “is emotionally closed off and has impulse control issues” to go first on the player’s turn merely by electing to do so. It allows the bookish cleric to watch the battle explode around him, and then figure out a way to save the rest of his party. It also gives him time to calculate what bloody spell to use.
What about structured events like death saving throws or effects that end the players turn? The effect ends at the end of their turn like usual. So, if you go first, the effects ends after your turn, not the end of your teams turn. You go last? You can eek out just a few more seconds of effect’s playtime if you really need it. Death saving throws are, by convention, done at the end of the teams turn, but I suppose the player can do it whenever. Luckily enough, my players don’t meta game enough to warrant an real concern.