indescribable 
professor has her masters degree in ancient Greece, so I knew how she'd react when I told her that I aimed to sum up everything important and relevant about Hellenic animal sacrifice in 6 to 10 paragraphs. I was right, too: shock, horror, and rolled eyes. 
The standard college textbook on ancient Greece, Walter Burkert's Greek Religion, dedicates about 20 pages to the subject, describes itself as only an overview, and makes no effort whatsoever to discuss what modern equivalent religious services might look like. If I didn't know that it was impossible, I wouldn't know to be intimidated at the effort. (Insert wry grin here.) But here I go, because there are four things that get thrown in your face any time you talk about Hellenic reconstructionism: blood sacrifice, sexism, slavery, and pedophilia. I know this, so I know that I can't talk about respecting the Greek gods and Hellenic religion of the classical era, in front of a modern mostly-American audience, without addressing these topics.Of course, here in America a lot of the hostility towards even the mention of animal sacrifice deflated to mere huffiness and feelings of superiority after the Supreme Court's 1993 ruling in the case Santeria Church of Lukumi Babalu Aye v. City of Hialeah, Florida. In Santeria v Hialeah, we had the case of a city that was grossed out by Santerian animal sacrifices, hoped specifically to shut down a specific Santerian church that had opened within the city, and so passed a law banning animal sacrifices. The law got overturned at the Supreme Court on the grounds that the city's deliberations showed that the intent of the law was to discriminate against a specific religion. Nonetheless, two other jurisdictions that had anti-animal-sacrifice laws accidentally misinterpreted this ruling as a blanket legalization of animal sacrifice, and so every jurisdiction in America takes it for granted now, ten years and more later, that animal sacrifice has been declared a basic religious right by the US Supreme Court.
Assuming that you have no problem with animals being killed for food, does it change how you feel about Greek temple sacrifices that the methods required were as humane and painless and non-terrifying to the animal as anything Temple Grandin might have come up with, subject only to the limitations of iron age technology? Actually, it probably won't soothe any of you, given the fact (incomprehensible to us post-moderns) that this humane and religious slaughter was performed as religious spectacle in front of tens, hundreds, or even thousands of people at a time. What could possibly have been the point of making men, women, and children watch while the poor animal is stunned and then bled quickly to death? Do those people benefit in any way from watching the butchers chop that animal up and serve certain organs to the honored guests raw? What's spiritual about watching the butchers carve the animal up, burn the fat and the bones and the hair from the hide on a sort of a forge, and boil the rest as a kind of barbecue or stew before handing out even portions to everybody else present? Am I saying that if we're going to save liberty and democracy in America, everybody has to go watch this sort of thing several times a month? And, well, ewww, right?
It would serve no purpose to restore temple animal sacrifice to 21st century Americans. That bugs me to say, because the gods are quite clear as to how much they wanted those sacrifices to take place. But I have a hard time believing that they'd still want it, because of two related "little" problems. First of all, those sacrifices wouldn't achieve the same goals. And secondly and even more importantly, there is absolutely no way that they could mean the same thing to a 21st century audience that they meant to an iron age Greek. When the ancient Greeks murdered an ox (and one of the Greek words for sacrifice means just that, "ox murder") they weren't just killing a food animal. They were destroying an important piece of the community. They were committing that classic upper class sin of "dipping into the capital." There isn't enough pasturage in ancient Greece to raise significant enough numbers of cattle to count on their hides for large amounts of leather, to count on their meat for any significant amount of the community's food. Nonetheless, every small farm needs at least one ox, and somebody needs to be raising more of them. The ox isn't primarily a future source of food, it's an absolutely essential piece of farm equipment. It turns inedible grass and stalks into motive power for plowing, heavy transportation of agricultural produce, fuel (dried dung) for starting fires, and nitrogen-rich fertilizer for the fields. Similarly, a goat or a sheep is only food in the richest or poorest of times; the rest of the time it's a garbage disposal that produces wool and milk. Even the lowly piglet is a garbage disposal that produces fertilizer. The situation of the ancient Greek, especially during the archaic dark age when the gods were here to instruct them, would be to us as if in order to eat any red meat, a farmer first had to burn his tractor.
Until the Greeks developed the technology of large-scale fishing fleets with nets, meat was something that they only ate at religious ceremonies. Superstitious Greeks convinced themselves that this was because the Gods needed the smoke from those fires for their nutrition. Poppycock. The gods needed or at least wanted the community to thrive, and that community could only thrive if to everybody it was a reflex that one only dug into savings, impoverished the farm, sacrificed long-term utility for a short-term meal, to serve the larger community. Red meat is never eaten privately. And during the limited time when fish was available in the marketplace for private consumption, the Greeks developed a special insult for someone who ate very much of it. An opsophagos is a special kind of a glutton, a glutton for luxuries, a meat addict, and the Greeks had a deep-seated suspicion of people being slaves to any appetite, whether for food or drugged liquors or sex. Another element of that contempt was for anyone who'd abuse his wealth to monopolize, or consume more than a fair share of, any commodity.And that brings up another part of the goal of the sacrifice ceremony, one that's basically impossible to meet in this manner by any society with freedom of religion. The larger sacrifices (such as the "hecatomb," which means the sacrifice of 100 oxen at a single ceremony, enough to feed a big percentage of the town at once) was to act as a kind of progressive income tax, at least in Athens. To qualify for certain privileges, you had to demonstrate that your farm and/or other businesses were generating annual produce in the highest of three ranges. However, being on that list made you eligible not only for those privileges, but for a lottery drawing by which the expenses of major city festivals were divvied up. And on top of those levied religious duties, every successful businessman and farmer knew what the religion taught about windfall profits. Whether you think you earned it through your own skill, or know that you received it in any part through the favor of the gods, if you didn't want to arouse the anger of the gods then your first duty was to buy up 100 oxen and offer them at a city temple.
All of these religious duties combined with various restrictive laws (for example, on land ownership) to try to prevent anyone from rising above a certain upper-middle-class level of wealth. When the gods were most active among the Greeks, the tended to vex and punish those who rose above that status. When society had become successful enough that the gods no longer had to be here in person for us, for a while men remembered to prevent that kind of concentration of wealth, because they knew that if you let enough men acquire vast wealth, sooner or later some of them will use that wealth as a lever, as a power tool to wreck democracy and freedom. They were right to fear this, too, as I'll show when I get around to talking about why this religion vanished (for a time) from the face of the earth.
But in the meantime, a society with freedom of religion can't realistically depend on religious law as a way of keeping the rich from becoming too powerful, or as a way of discouraging short-term thinking among the middle class, or as a way of teaching society that certain luxuries are only appropriately enjoyed if you share them with others. And that presents me with a terrible dilemma, because that would leave us with only teaching and debate and philosophy and politics as tools with which to achieve these worthy goals, and those are are slender reeds that may not be up to the job. But then, if you get down to it, I suppose that neither was religious law for them.
As an aside, while we're talking about religious observance, I wrote an essay quite a while ago about ancient Greek prayer. You might find it interesting.