As for how I do initiative, and my thoughts behind it:
First off, I really dislike cyclic initiative where particpiants act in the same order every round; as it does nothing to replicate the chaos of battle and also leads to far too many meta-game decisions being made based on the turn order, which the characters wouldn't know. So, it's always going to be re-roll every round.
Second, I try to avoid stat-based modifiers of any kind on initiative; as that stat - particularly in a re-roll system - immediately becomes the most important stat in the game.
Third, I don't mind simultaneous initiatives; as in reality things can and do happen at the same time.
Fourth, every action (excluding reaction-type things e.g. AoOs) from every participant gets its own independent initiative. Yes, this means if you have, say, multiple attacks in a round you roll a separate initiative for each one to determine when they occur.
Fifth, in a re-roll system with (sometimes) many participants, using d20 for initiative is cumbersome in various ways not least of which being I-as-DM don't own enough d20s to roll and leave in front of me for all the monsters; never mind I need one or two for rolling attacks, saves, etc. However, we all own loads of d6s (if not, they're dime-a-dozen to pick up at the dollar store). D6s are also convenient in that you can leave them on the table in front of you once rolled and they're way less likely to get knocked over than is a d20.
Sixth, spells take time to cast; and this is reflected by you starting to cast on your rolled initiative and resolving a certain number of initiative counts (i.e. segments) later, with this time amount set by the specific spell you're casting.
So, d6 for each action or attack, re-rolled every round.
There's a few quirks and modifiers we've developed over time, but none are essential.