Method for playing an online text-based RPG that doesn’t require everyone to be online at once?

I am contemplating starting up a web-based text game for my D&D group to be played alongside our weekly sessions. The idea would be to play through events related to the main storyline, but not so involved that the outcomes would directly affect our current adventure.

I don’t want to require everyone to be online at once (or else this is just another session), so IRC and chatrooms are out. I am looking for a method that allows us to play at a slow pace but gives everyone a chance to respond to what the GM says before moving onward with the story or the combat.

The only idea that comes to mind is a forum-based game with heavy restrictions (e.g. each player must post a response or opt out of responding before the GM posts again). For combats I would post maps with the position of all the monsters and characters between each turn.

Does anyone have any experience running a game like this?

What’s the best way to run a web-based text RPG in such a way that not all players need to be online at the same time?