Basic Hellenic Offering Ritual
Sourced from here. My own notes have been added in italics. Also note that this is just a >guideline<. Any step can be changed or omitted as needed.
The key elements of a bloodless offering are as follows:
Procession (Pompe)
The participants move from a central gathering place into the sanctuary and up to the altar.
(Most modern altars will be in the home, so the procession doesn’t have to be very long. Depending on where your altar/shrine is situated you could start from outside and process inside, start from the next room, or upstairs/downstairs. You could even just walk around the house before returning to the altar. Side note – If you lack space, rituals can be done at a shrine, a separate altar does not need to be set up unless you feel the need.)
Marking the Space
The offerings and sacrificial implements are carried around the altar. There is a good deal of controversy about whether this should happen in a sunwise direction or not; as there is no consensus within the community on this issue and no incontrovertible historical evidence one way or the other, this detail can be left to the discretion of the ritual leader or group, or omitted altogether.
(My shrine is a window sill, so walking around it isn’t possible. I usually skip this step.)
Preparations for the Sacrifice Proper
Purification
In Greek, this phase of the ritual is referred to as arkhestai, meaning “to begin.” The hands of the participants are washed with lustral water (khernips). The water may be purified by plunging a piece of smouldering wood from the fire into it or salt water may be used. The water is also sprinkled over the sacrifices and altar.
(For my khernips I use plain tap water in which I extinguish a lit match or for special occasions a smouldering bay leaf. I also mix in some sea salt.)
Preliminary Offering of Barley and Prayer of Invocation (katarkhesthai)
Participants take handfuls of barely groats (oulai, oulokhutai). The sacrificer, with uplifted arms, speaks a prayer or other call to the gods. The barley is then thrown onto the altar.
(It is entirely too messy to toss loose barley at my shrine, so this offering would be fine in a bowl. You could just run your fingers through the barley and let it trickle back into the bowl while speaking the prayer.)
Final Preliminary Offering (aparkhesthai)
Using the sacrificial knife, the sacrificer cuts off a small portion of the offering and casts it into the fire. With animals, a bit of the hair from the animal’s forehead was used; a small piece of bloodless food offerings can be cut or broken off.
(Uses of bowls or plates at this stage is fine. If you’re lucky enough to be able to have an open flame big enough to burn things in then I envy you.)
The Sacrificial Moment and Ululation (he thusia kai ololuge)
The offering is cut, broken, or spilled out on the altar. At the same time, the women give a shrill trilling cry.
(If you live in an environment where shouting would draw attention or get you some funny reactions then you can skip this step or do something a little quieter.)
Distribution of the Sacrifice
A small (often inedible) portion of the offering is set aside solely for the gods; this is usually burned (for the Olympians) or placed in a pit in the earth (for the Khthonioi). Some sources indicate that the first portion of the sacrifice was offered to Hestia, and this may be incorporated into the order of the ritual. Small offerings may then be made to each deity associated with the festival. The rest is cooked (if necessary) and distributed among the people. Today this last act may take place at the concluding feast rather than at this moment.
(It is personal choice on how much of the offering you share.)
Libations
Offerings of wine, milk, oil, or other liquids are poured out onto the fire. In modern usage, potable offerings may be poured into a bowl from which the participant then sips. The remaining contents of the bowl are then poured out onto the fire or the ground.
(Offerings and libations can be left on a shrine/altar for as long as you like but be aware of rotting and evaporation.)
Conclusion and Call to the Feast
The offerings (and other food) are now shared by the participants. Traditionally all the food was consumed at the ritual site. Modern usage allows for excess food to be taken away or donated to a food bank or other charitable cause.
(I usually take the leftovers and leave them outside for the deity/deities that I’ve been honouring. If leaving foodstuffs outside is not a viable option for you, you can dispose of them in the trash, the gods will understand.)