I'm not sure if it's been mentioned, but there could be more social effects of religion. Pious dwarves getting happy thoughts from prayer has been mentioned, but what about funerals?
With a temple as part of your fortress and a religious noble (i.e. a clergyman brought along by a noble as part of their court) you could hold funeral services for dead dwarves. This would prevent other dwarves getting bad thoughts due to their death and would give them good thoughts, due to the assurance that their loved one has passed on in the correct manner.
Similarly you could have the temple schedule sermons a certain number of times every month. Based on a dwarf's piety they either would or would not attend the sermon, and those that did would gain happy thoughts. You could also have emergency sermons that you could call during times of intense unrest (e.g. in the middle of a tantrum spiral) which would make the pious dwarves drop everything (both figuratively and literally, knowing Dwarf Fortress) and attend, giving them happy thoughts and possibly averting a tantrum spiral.
There's also the crossover between secular government and religion to consider. Clergymen, being the voice of the gods assembled, could alter or overturn noble's mandates, or set their own with religious goals in mind, e.g. "Make 10 holy symbols", "Build a lavish temple" or, as has been mentioned before, sacrifices like "Sacrifice two bulls".
Another idea could be to have certain artefacts (such as those made from "possessed" strange moods) be considered "holy", in which case they could be put on display in a temple, causing pilgrims to make journeys to your fortress where they would then pay a toll, either in goods or money, to view the holy item.
There are lots of possibilities and not all of them will be implemented, but there's more to religion than just gods; there are those that claim to represent them, too.