This is another idea I came across on a forum, and which got stuck in my head long after reading it. If you are reading this, and you happen to be the person who came up with it, leave a comment and I'll add your name here.
The idea was to basically stop rolling for initiative in most cases. Here's how that might work:
The DM has an idea of what each monster will do, and the players declare their actions as normal.
After that, the DM basically adjudicates the order things happen in, with as few rolls as needed. Some considerations to make:
Characters with ready missile weapons (either already loaded, or the launcher in hand and ammunition fairly ready) would generally be able to attack before other things occur.
Spell casting can occur when the DM feels it's appropriate, and how he generally feels about spells. I tend towards feeling that spell casting takes up most of the combat round, with spells taking effect after any "quick actions".
Characters moving around and battling in melee likely won't need initiative rolls in most cases either, unless exact positioning is important. Let attacks occur simultaneously. If a "round" is multiple blows, its entirely possible that combatants could injure each other, or even kill each other (the fighter strikes down the orc, only to succumb to his wounds).
Where movement is involved, rather than making it all or nothing (as with initiative rolls), have characters meet at the halfway point (effectively prorating the movement) or similar.
In cases, where it's important whether this or that action occurs first, a simple D6 roll can be used (high rolls favour heroes, low rolls favour monsters) but try to avoid that as much as possible.
The first few times you try this, it'll be awkward and a bit slow, but once you get the hang of it, it'll fly and you likely never even notice that you're missing the dice throws.
Do you have a non-standard way of handling turn sequencing and initiative? Let us know in the comments