My table does a group initiative. Though it kind of works like an Initiative check for a bonus turn.
At the beginning of combat the PCs roll initiative against a single monster Initiative roll (although you could also set the DC to a static 10 or 15 to speed things up even more).
The PCs who beat the monster's Initiative get to act first. But they get to act in whatever order they want, as they choose regardless of who rolled higher.
Then all the monsters go. And that's essentially the end of the round.
In each subsequent round, the players go first, again acting in whatever order they choose regardless of what they rolled or what order they acted in the previous rounds.
Where this speeds things up is that the players are in complete control of who goes next, and we tend to pay far more attention to what the others are doing, looking for a good time to take our own turn. Plus, if there's a player hemming and hawing about what to do, or looking up a spell, someone else can step in and do their thing in the meantime. It kind of bundles all the indecision together, and that helps, too.
A round can go by so quickly, and being withing the control of the players, our DM often doesn't realize that it's the monsters turn again, and we're sitting around waiting for him to act until one of us says "uh, we're done. It's your turn."