Hello, Amethystine of Darnassus, with my experiences and input on healing heroic Icecrown Citadel 10s on a Holy Priest.
I came into the guild run 4/10, so we went straight into the Plaguewerks... though I can still offer some advice for Holy Priests for one of them. Lootship naturally doesn't -need- tips, and alas I've not yet gotten to try LDW, or managed to down DBS.
Lord Marrowgar: I would strongly suggest starting your run with either Chakra: Renew or Chakra: Heal. Your double HoT in Renew is great for helping smooth out spike damage, and the reduced cooldown of Renew in its Chakra is awesome for healing Bone Spikes, keeping tankier healers on the tanks. Another strong heal to use is Lightwell, and with practice, the Bone Spiked victims can use it to heal themselves, freeing up every bit of healing to focus solely on the tanks. Remind your guildies/friends in the run that Lightwell can be clicked while incapped, stunned and even mid-spellcast.
During Bone Storm, your Chakra should have renewed itself, and your mana should still be pretty high. Switch to Chakra: Prayer and lay down Sanctuaries on either/both the left and right side of the raid. Drop a Prayer per group, keep Mending going around, and ready your Flash Heals for Bone Spikes.
Bone Storm over (and nobody should be dead, hopefully), rinse and repeat.
Festergut: This was another good fight to practice switching Chakra during the battle. I started in Prayer of Healing, and was with the tanks and other healers beneath the boss. With Prayer of Mending bouncing between us, I was largely able to simply drop Sanctuary on both ourselves and, during spore, the collapsed DPS. Prayer of Healing easily kept up with the blight damage, and heals were used between Prayers along with Circle to keep people more topped off.
Right around the tank switch, or even a little before then, you'll be raid healing less, and tank healing much, much more. I used Chakra: Heal, spamming Heal to keep the tank well during the 'third phase' of Blight, while popping up Holy Word whenever it was up. As he goes into his last, high-output tanking phase, I naturally pop Guardian Spirit, and still spam Heal, though now whenever it was up, I switched to Holy Word > Flash Heal > Greater Heal whenever Holy Word was up. Renew should be permanently on the tank, due to your Heal spam, and I was never in danger of running out of mana.
Rotface: An easy one, just employ Chakra Renew, using Holy Word as well whenever someone gets the Injection. Save Guardian Spirit for the rare instances that a tank may get nicked by whomever has Vile Gas. You shouldn't even have to drop a Shadowfiend.
Professor: This one was fun for us, as we actually hadn't tried it before. I didn't strictly heal this fight, in fact opting to switch to Shadow for the added DPS while retaining some minor healing functionality. Like Rotface, I imagine Renew being the most useful Chakra for this battle, easily able to keep up the Orange kiter, the Green target, and keeps our heals from being too shocked by a stray Malleable Goo (it happens, so don't panic). For strategy, we stuck our position to just behind where the Green Ooze spawns. Typically, we were able to kill it after the first knockback without it getting a second (even after a bad transition to third phase, in which he double spawned -after- the orange came out) The Orange Ooze was neatly kited through the group, dying before it reached its target. Not a difficult fight by any stretch. Just remember to take care who's getting Unbound Plague.
Blood Princes: Healing this fight in heroic is largely unchanged for us. I preferred Renew yet again to help you move further without getting dangerously low on health, though I switched to Heal whenever Keleseth was active. Guardian Spirit's best used if you think an Empowered Flame Orb isn't drained enough for its target to survive otherwise, along with a bubble for good measure. This fight's still more about raid awareness and kinetic bomb/vortex attention than anything. The only problem you might run into is healing range, so remember, you can run a goodly while. Just make sure you -stop- now and then to let Shadow Prison wear off.
Blood Queen: Another fight where we opted to two heal, switching my Holy spec for Shadow... and this is a very, very good approach to heroic Blood Queen. The DPS held back just a little, enough for me to get the first bite. Popped Fear Ward on the pull, and Glyphed Mass Dispel for the air phase are both very powerful tools for any spec. As a Holy Priest, I strongly recommend Chakra: Prayer. You can use it intermittently to keep the Chakra going, Heal spam to keep people up, and on the Air Phase, you can drop an initial Prayer, and start up Divine Hymn as it lands. DPS should be high enough now that you can kill her before the 2nd Air Phase, though your biters should still remember to hold off their Bite until about 5-6 seconds left.
Dreamwalker: We didn't -quite- get it down, sadly, but came within 3% of it. Holy Priests are one of the healers I strongly recommend going into Portals. Chakra: Renew and Holy Word can both be kept up on Dreamwalker, and Renew shot aside onto tanks helps a lot. We can heal -very- quickly, using our infinite Flash Heals followed by a sped-up Greater Heal ala Serendipity. So long as you don't miss Portals, you'll likely contribute more overall healing on Dreamwalker than other healers can... and during Heroism/Bloodlust/Time Leap, which I recommend hitting after the portal following 75%, drop Flash Heal altogether and spam Greater Heal instead for the increased throughput.