Once you have reached "Diviner" (Rank 7) and have finished all of Endryin Llethan's quests, he will send you to Archcannon Saryoni (of "Saryoni's Sermons" fame) for further duties and will give you a key for the locked door in his office.
If you have been talking to people about Morrowind Lore and "little secrets", you've no doubt discovered that the locals lump the Daedra into two groups: good Deadra and bad Daedra. The bad Daedra are known as the "Four Corners of the House of Troubles". You need to go visit the shrines of the four bad daedra, starting with Malacath, and report back to Saryoni after each one. Two of the shrines require offerings and two do not. Just activate the statue to invoke the blessing. You'll receive a long-duration attribute or skill boost and +1 to your overall reputation after finishing each.
Malacath
Malacath's shrine is kind of southwest of Dagon Fel. Just follow the road, take the first right and turn right again when you see the strapping Orc fellow standing by the side of the road. That road will take you past four more Orcs (all heavily armored) and you'll find Malacath's shrine at the end of that road. The four Orcs will probably decide that they don't like you very much. Whatever you do, don't let them dog-pile you. Take them one at a time. When you're done, activate the shrine with the four Daedra Hearts in your inventory to complete the pilgrimage. The shrine's 2-day blessing of Strength (10-20 points) might come in handy for the load of their goodies. Ocrish Armor and Weapons are hefty, but it worth it when you're cash-strapped.
Mehrunes Dagon
Dagon's shrine is at Ald Sotha. You've already visited it for the Temple, and may have cleaned it out for the Morag Tong. It's northeast of Vivec. If you cleaned it out, then the only things you might have to deal with are respawning Daedra. You don't need to donate anything, so if you can run in, activate the statue (no donation required), and run back out again, you might pull it off without a scratch. You'll get a day-long blessing that fortifies your Destruction, Axe and Attack. Report back to Saryoni.
Molag Bal
Bal Ur is just upriver from Suran. You might have visited it for a cure for vampirism if you took a walk on the dark side. Aside from that, it is not the destination of any other quests. It's just over the mountains north of Suran, but there is no road leadint to it. You'll either need to swim around the headland, levitate over the mountains or make a somewhat lengthy trek north on the west side of the river, cross and come back down the east side.
There are a few crittes that wander the surface ruins, but nothing that you wouldn't normally be able to handle. Aside from a couple of worshippers, you'll run into your biggest problem in the shrine itself because you have to cross a very narrow, twisty, rock bridge to get from the tunnel to the shrine. In the meantime, Derar Hlervu (a Level 27 Witch) will be sending stuff your way. Although she has a good selection of spells (like Bound Longbow), she doesn't have any ammunition and no ranged spells. She can, however, summon a few critters. This makes it possible for you to be knocked off of that narrow bridge into the lava below (major owie). So if you can levitate over to the shrine, your life will be a bit simpler. She has an Ebony Mace, reasonable skill and 191 Health, so she's not a complete push-over. You'll find a random potion and magic item on her when you're done.
Activate the shrine to get a day-long Speechcraft boost and a 30-second Command Humanoid (5-25 points) spell. Report back to Saryoni.
Sheogoath
Sheogorath is the other shrine that needs an offering. But not something normal. You must offer a "Gambolpuddy". Sounds like cat game ("gambol" as in playing or frolicking and "puddy" as in "I tawt I taw a puddy-tat"). Anyway, Saryoni tells you that it's somewhere in the Ald Daedroth Shrine and you'll need to find it to make the offering.
Getting to Ald Daedroth can be easy or difficult, depending on whether you've been named Nerevarine by the Ahemussa Ashlander tribe. If you have, then you've probably already cleared out the shrine, so it's just a matter of getting the Gambolpuddy and activating the shrine. If you haven't, then you'll need to clear out the shrine (sort of).
The Ahemussa Camp is almost directly north of Tel Vos. Ald Daedroth is almost direclty north from there. Swim, Levitate or Water Walk. The ruins and the entrance are on the northwest side of the island.
When you enter the Outer Shrine, there are two Sheogorath worshippers who will attack you. With the entrance behind you, to your left is the "Left Wing", to your right is the "Right Wing" (makes sense) and directly in front of you is the Inner Shrine. You do not need to mess with either the left or right sections, but there are some decent goodies if you're a little strapped for cash (several Ebony Demon Maces in both sections) and the Right Wing also has two Sanguine items (Green Wisdom and Smiting) for the Morag Tong's "Threads of the Webspinner" quest.
If you do not feel like messing with those sections, go directly to the Inner Shrine. There are a Sheogorath worshipper (Eigma) and a Temple Ordinator (Nels Indrano), who will hopefully kill each other before you show up on the scene, leaving only a Golden Saint at the rear of the section to deal with. At the rear of the Inner Shrine is the entrance to the Antechamber section. Go in. You'll run into a 3-on-1 fight between Sheogorath worshippers and an Ordinator (I've never seen the Ordinator win on his own). You'll find a few enchanted weapons and Red Despair Amulets (5-second targeted Absorb Willpower 4-8 points) when it's all over. There is another wandering Golden Saint, too.
Go upstairs to find a small party going on. A non-hostile Scamp and a couple of Sheogorath worshippers, including an Orc who thinks he's a cat (admire his tail to boost his disposition). There is also some Moon Sugar on a pedestal which, if you give it to the Orc, will get him to tell you that he hid something you need under a pillow. Since there are only two pillows in the room, it's pretty easy to find the Gambolpuddy.
Take it back to the Inner Shrine and activate Sheogorath's statue. Your "blessing" (Blessing of the Fourth Corner) is an all-day Fortify Intelligence/Luck/Personality/Willpower (1-25 each) and Drain Intelligence/Luck/Personality/Willpower (1-25 each). This can result in a net drain on any or all of those four attributes. Sheogorath giveth and Sheogorath taketh away. But you get a 25-point faction rep boost with the Temple for finishing off all four shrines and +1 to your overall reputation, so it's not a complete loss (and the spell will wear off eventually).
Report back to Saryoni and he will tell you that he has one more thing that you need to do before he will promote you to Patriarch.
This is the last quest for the Temple. You need to go to Mount Assarnibibi and pray at the shrine to receive the Ebony Mail. Getting there is the hardest part. Aside from a random wilderness critter or two, there isn't anything at the shrine itself.
Mount Assarnibibi is north-northeast of Molag Mar. If you don't feel like levitating (Saryoni marks it on your map, so it's not hard to find), go east/northeast in the foyada that runs by the Silt Strider platform. Turn right at about the point where the foyada turns north. Turn left when you get to the Helan Ancestral Tomb, bear left at the fork in the trail and stop when you get to the shrine. It's surrounded by a bunch of upright rocks, so the shrine is a little difficult to see, but the circle of big pointy rocks is easy enough to find.
The shrine will cure any diseases or poisons and will add the Ebony Mail to your inventory (Constant Effect 75% Resist Fire, 50-point Shield and 20% Resist Magicka). At 20 pounds, it counts as Medium Armor (woo-hoo). You also get +1 to your overall reputation.
When you report back to Saryoni, he tells you that you can keep the Ebony Mail, promotes you to Patriarch and essentially retires from the Temple. He stays in his office, however, since you will need him to get to Vivec for the Main Quest.