Haven't had a chance to try it yet, but I'm going to go with side initiative, rolled every round, and using the following combat sequence, modified from Moldvay:
1. Morale
2. Movement (Includes Dash, Disengage, Dodge, Help, Hide, Search, and Use an Object Actions)
3. Missiles/Movement (Attacks with ranged weapons, plus any movement those characters have left over)
4. Magic/Movement/Saves (Cast a Spell, plus any movement those characters have left over, plus any saves that need to be made)
5. Melee/Movement (Attacks with melee weapons, plus any movement those characters have left over)
Reactions can occur whenever triggered.
What I want to do is get that same feel as in Classic D&D, where you didn't have each person do their full turn while the others waited, but the turns were more woven together.