Theophrastos, as quoted in Porphyry’s On Abstinence from Animal Foods 2.24
There are, moreover, three reasons altogether for sacrificing to the Gods: to honor them, to give thanks, or from need of some thing. We ought to offer the Gods the first-fruits of all we receive, for it is their generosity that makes our living possible. Further we honor the Gods because we want evil to be averted from us and those we love or for an increase of good things, or out of gratitude because they have benefited us in the past or simply to honor their condition of goodness.

Porphyry, On Abstinence from Animal Food 2.16
He thought fit to ask him after what manner he reverenced the Gods. Clearchus answered him that he diligently sacrificed to them at proper times in every month at the new moon, crowning and adorning the statues of Hermes and Hekate, and the other sacred images which were left to us by our ancestors, and that he also honored the Gods with frankincense, and sacred wafers and cakes. He likewise said, that he performed public sacrifices annually, omitting no festive day; and that in these festivals he worshiped the Gods, not by slaying oxen, nor by cutting victims into fragments, but that he sacrificed whatever he might casually meet with, sedulously offering the first-fruits to the Gods of all the vegetable productions of the seasons, and of all the fruits with which he was supplied. He added, that some of these he placed before the statues of the Gods, but that he burnt others on their altars.

Sallustius, On the Gods and the World 16
I think it well to add some remarks about sacrifices. In the first place, since we have received everything from the Gods, and it is right to pay the giver some tithe of his gifts, we pay such a tithe of possessions in votive offering, of bodies in gifts of (hair and) adornment, and of life in sacrifices. Then secondly, prayers without sacrifices are only words, with sacrifices they are live words; the word gives meaning to the life, while the life animates the word. Thirdly, the happiness of every object is its own perfection; and perfection for each is communion with its own cause. For this reason we pray for communion with the Gods. Since, therefore, the first life is the life of the Gods, but human life is also life of a kind, and human life wishes for communion with divine life, a mean term is needed. For things very far apart cannot have communion without a mean term, and the mean term must be like the things joined; therefore the mean term between life and life must be life. That is why men sacrifice animals; only the rich do so now, but in old days everybody did, and that not indiscriminately, but giving the suitable offerings to each God together with a great deal of other worship.

Epicurus, as quoted in Philodemos, On Piety 31
Let us sacrifice to the Gods devoutly and fittingly on the proper days, and let us appropriately perform all the acts of worship as the laws stipulate, in no way disturbing ourselves with opinions on matters concerning the most excellent and august beings. Moreover let us sacrifice justly for we draw closest to Zeus when we most act like Zeus.

Areios Didymos, Epitome of Stoic Ethics 3.604-3.662
The Stoics say that only the wise man can be a priest, while no worthless person can be one. For the priest needs to be experienced in the laws concerning sacrifices, prayers, purifications, foundations, and the like. In addition to this he needs ritual, piety, and experience in the service of the Gods, and to be close to the divine nature. Not one of these things belongs to the worthless; hence, also all the stupid are impious. For impiety as a vice is ignorance of the service of the Gods, while piety is knowledge of that divine service. Likewise they say that the worthless are not holy. For holiness is described as justice with respect to the Gods. The worthless transgress many of the just customs pertaining to the Gods, on account of which they are unholy, impure, unclean, defiled and barred from festive rites. For carrying out festive rites is, they say, the mark of a civilized man, since a festival is a time when one ought to be concerned with the divine for the sake of honor and appropriate celebration. So the person who carries out festive rites needs to have humbly entered with piety into this post.

Porphyry, On Abstinence from Animal Foods 2.37.4-5
There is a multitude of divinities which some call indiscriminately Gods and others more appropriately daimones. People have given some of them names, and they receive from everyone honors equal to the celestial bodies, as well as their own distinct forms of worship. Others have no name at all in most places, but acquire a name and cult inconspicuously from a few people in villages or some cities. There is a widespread conviction about this multitude of daimones, that they can do harm if they are angered by being neglected and fail to receive their accustomed worship, and on the other hand that they can do good to those who make them well-disposed by prayer and supplication and sacrifices and the shedding of blood and all that goes with it.

Lucian, On Sacrifices 2
So nothing, it seems, that the Gods do is done without compensation. They sell men their blessings, and one can buy from them health, it may be, for a calf, wealth for four oxen, a royal throne for a hundred, a safe return from Troy to Pylos for nine bulls, and a fair voyage from Aulis to Troy for a king’s daughter! Hecuba, you know, purchased temporary immunity for Troy from Athena for twelve oxen and a frock. One may imagine, too, that they have many things on sale for the price of a cock or a wreath or nothing more than incense.

Philodemos, On Piety 25-28
Therefore I think it is especially necessary to despise those who transgress or mock the traditional rites. Furthermore it will appear that Epicurus loyally observed all the forms of worship and enjoined upon his friends to observe them, and not just be in accordance with the laws. For as he says to pray is right and natural for man, not because the Gods would be hostile if we did not pray, but the act of doing so helps us gain a better understanding of those who surpass us in their power and excellence, enabling us to fulfill our potential. He also said that every wise man holds pure and holy thoughts about the divine, namely that the nature of divinity is great and august. And it is particularly at festivals that we attain our greatest understanding of things for during a festival all that a man can think about, and all that is upon his lips, are holy matters. He didn’t just advise others to participate in the worship of the Gods – indeed, he was very active in religious matters, sharing in all festivals and sacrifices, and especially the Khoes festival and the mysteries celebrated in his city and elsewhere.

Marcianus, Institutes 3.2-3
Things which are sacred, religious, and holy are not the property of anyone. Sacred things are those which are publicly and not privately consecrated; and hence if anyone should make anything sacred for himself privately, it is not sacred but profane; where, however, a temple has once been made sacred the place still remains so, even after the edifice has been demolished.

Lucian, On Sacrifices 10-13
That is the way the Gods live, and as a result, the practices of men in the matter of divine worship are harmonious and consistent with all that. First they fenced off groves, dedicated mountains, consecrated birds and assigned plants to each God. Then they divided them up, and now worship them by nations and claim them as fellow-countrymen ; the Delphians claim Apollo, and so do the Delians, the Athenians Athena (in fact, she proves her kinship by her name), the Argives Hera, the Mygdonians Rhea, the Paphians Aphrodite. As for the Cretans, they not only say that Zeus was born and brought up among them, but even point out his tomb. We were mis­taken all this while, then, in thinking that thunder and rain and everything else comes from Zeus ; if we had but known it, he has been dead and buried in Crete this long time! Then too they erect temples, in order that the Gods may not be houseless and hearthless, of course; and they fashion images in their likeness, sending for a Praxiteles or a Polycleitus or a Phidias, who have caught sight of them somewhere and represent Zeus as a bearded man, Apollo as a perennial boy, Hermes with his first moustache, Poseidon with sea-blue hair and Athena with green eyes ! In spite of all, those who enter the temple think that what they behold is not now ivory from India nor gold mined in Thrace, but the very son of Cronus and Rhea, transported to earth by Phidias and bidden to be overlord of de­serted Pisa, thinking himself lucky if he gets a sacrifice once in four long years as an incident to the Olympic games. When they have established altars and formulae and lustral rites, they present their sacrifices, the farmer an ox from the plough, the shepherd a lamb, the goatherd a goat, someone else incense or a cake ; the poor man, however, propitiates the God by just kissing his own hand. But those who offer victims (to come back to them) deck the animal with gar­lands, after finding out far in advance whether it is perfect or not, in order that they may not kill some­thing that is of no use to them; then they bring it to the altar and slaughter it under the God’s eyes, while it bellows plaintively—making, we must suppose, auspicious sounds, and fluting low music to accom­pany the sacrifice! Who would not suppose that the Gods like to see all this ? And although the notice says that no one is to be allowed within the holy-water who has not clean hands, the priest himself stands there all bloody, just like the Cyclops of old, cutting up the victim, removing the entrails, plucking out the heart, pouring the blood about the altar, and doing everything possible in the way of piety. To crown it all, he lights a fire and puts upon it the goat, skin and all, and the sheep, wool and all; and the smoke, divine and holy, mounts upward and gradually dissipates into Heaven itself.