Yes, I've now done I believe over half of the specs for Legion at this point, but no healers. Well, the main reason for that is that I don't really play healers very much. My second Monk character is a Mistweaver, and has made it up into Pandaria, so I guess I can talk a bit about them, but I've never done any serious healing in heroic dungeons or raids.
Two massive changes are coming to Mistweaver Monks. The first is arguably less profound - Chi is becoming a Windwalker-only mechanic. Mistweavers will simply cast their spells for mana like other healers. The other huge change is that Crane Stance and all damage-for-healing functionality is going away. This is a pretty big disappointment to me, as it's the main thing that got me to try the spec in the first place. While Discipline's Atonement mechanics are remaining (and becoming the core of that spec,) Mistweavers are losing their "Fistweaving." What's surprising about this is that Mistweavers were built from the very beginning to incorporate this, and I'd argue that it was the defining feature of the spec, even if there were traditional healing options.
But I suppose one reason they want to play conservative (or more accurately, reactionary) with Mistweaver is that if Discipline winds up being a disaster, Priests can always respec to Holy if they still want to heal. Monks only have the one healing spec, and Blizzard wants to make sure that it works properly, even if that's at the expense of spec/class identity.
So let's look at your heals.
Effuse is a fast and efficient heal with a 1-second cast time.
Enveloping Mists has a 2-second cast time, healing the target over 6 seconds and increasing your healing to them by 30%.
Soothing Mists is now a passive. When you heal a target with Vivify, Effuse, or Enveloping Mists, you trigger Soothing Mists on the target. You'll automatically continue to heal them every 0.5 seconds until you take another action. Essentially, this is your "filler heal," but does not require you to hit any buttons or take any mana whatsoever.
Renewing Mists is an instant-cast heal on an 8-second cooldown that heals the target over 20 seconds. If it overheals, it will travel to another target within 20 yards. Each time it heals, it has a 4% chance to increase the healing of your next Vivify by 50%.
Vivify is a 1.5 second cast heal that heals the target and two nearby injured allies.
Thunder Focus Tea is a 45-second cooldown affects your next healing spell. Renewing Mist doesn't trigger a cooldown. Enveloping Mists is instant-cast. Effuse does 200% more healing. Essence Font can be cast while moving. And Vivify becomes free.
Essence Font is a channeled spell over 3 seconds that sends out bolts of healing to up to 6 allies within 25 yards, healing them for a bit and then putting a HoT on them for 6 seconds.
Revival is a 3-minute cooldown that heals party and raid members within 40 yards. for a huge amount and clears them of magic, disease, and poison effects.
Life Cocoon is a 2-minute cooldown, putting an absorb shield on the target that increases the periodic healing they take by 50% for up to 12 seconds.
Mastery: Gust of Wind causes your targeted heals to also give the target a burst of healing.
In addition to all of these, there are a number of melee attacks to allow Mistweavers to solo, like Tiger Palm, Blackout Kick, Spinning Crane Kick, and Rising Sun Kick. These do not affect healing in any way.
Obviously, healers don't really have a "rotation" per se. Enveloping Mists looks like the most expensive heal at 6% mana, and will probably be mostly for emergencies. Likewise, Vivify costs a fair amount as well at 4% mana, so you'll want to use that for situations where there's high raid damage (as this is kind of a "cleave" heal.) For light damage situations, you'll probably just want to keep Renewing Mists floating around and Effuse targets if they're in trouble, but then let Soothing Mists keep the tank up.