Druid Healer / Shaman Healer
I fail at the new dugeons so I can't give a great perspective of that.
On my highest geared team (5400gs average) I can only kill the first boss in BRC, trash and other things are no problem and the problem isn't healing it's mechanics and me on a slow learning curve. This is my druid healer team. This team is currently 81.25.
On my lowest geared team (4000gs average) I can't kill the first boss in BRC, but have no problems with the trash and can complete all group quests easy. This is my shaman healer team. This team is currently 82.3.
To heal effectively I think you just have to use what suits your style. Personally I think druids are the best healers, but many say that discipline priests are the best healers. Everything is always situaltional though.
My two guildmates with similar mixed teams have no problems with BRC, so I know it's just me that is being unskilled and unwilling to adapt (yet).