I’m going to be sharing some of my own rituals that I’ve prepared to celebrate festivals honoring the Greek gods; some will be modern versions of festivals known in the ancient world, others will be new, and I’ll clear about which are which.
Before that, though, I thought I should say a bit about the ritual format I use and why.
The rites are written to be used by a single person or a small group and might require some adaptation for a larger group. Please feel free to use them for inspiration, to make changes to suit your situation, or to follow the script as is. (I personally like to have a script ready when I perform a ritual although I may or may not stick with it throughout the rite. I know others who work strictly off the cuff, which I admire greatly but cannot count on myself to do well every time!) There’s a consistency among the different rites I do and some portions (particularly the framework) tend to stay the same because I like to maintain a cohesive liturgy–it puts me in a ritual frame of mind. 
I try to use a format based on what we know of ancient practice, and I usually try to keep it simple (again a matter of personal preference). Please note that what follows is a description of how I, myself, do ritual; other Hellenic polytheists may and do use different methods and formats, and you may well see other rites with different or additional steps depending on how the rite was interpreted.
Procession
Historically, most festival rites began with an elaborate procession, ending at the ritual site. Typically such a procession would be a parade through the city; while this is not an option for most of us, the significance of the procession remains, as a period of transition into a sacred space from a mundane one. I sometimes “process” by consciously taking a few steps and standing before the altar, thus moving from outside to inside a sacred space and a sacred state of mind.
Purification of participants
This stage of purification is accomplished with lustral or holy water. You’ll need a bowl of clean water and a stick to light (a thin piece of wood or an incense stick will work). To create the lustral water, light the stick and extinguish it in the clean water, saying “May all who wash in this water be made pure!”
After the water is ready, all participants lightly wash their hands and face with it. With a medium-to-large group I’ve found it works well for the officiant to use a ladle to pour water into participants’ hands over the bowl.
Purification of ritual space
The purification of space is accomplished with barley. You’ll need a small bowl of uncooked barley grain (either hulled or pearl barley is fine). Pass the bowl among the participants so that each can take a small handful of the grain.
To purify the altar space, say “May all that is impure be gone from here!” and throw the barley on the altar. If you’re indoors you may want to do this carefully; if you’re lucky enough to be using an outdoor altar, feel free to hurl the barley with great abandon!
Invocation
A call to the deity or deities being honored in the ritual. (The term “invocation” is used here in the liturgical sense rather than the magical. :)) This serves both to get the deity’s attention and as an invitation to the deity to join the rite.
Statement of purpose
A reminder of why we’re all here and what we are celebrating. Even if it’s just me, I include this step as a further affirmation of purpose.
Actions specific to the ritual
Different festivals will have different things going on at this point. For example, at the Khalkeia, a festival for Hephaistos and Athena, I include a blessing of crafting implements.
Libations
Libations are made with any drink appropriate for the occasion and the deity or deities being honored; wine is common and traditional, but other beverages such as juice are fine as well. If the rite is to honor a chthonic spirit, a libation of milk or of honeyed water might be appropriate.
In order, libations are usually made first to Hestia, then to the deity or deities being honored for the festival, then again to Hestia (who receives her due first and last). Finally, once the gods have received their libation, the cup is (usually but not always–for example, in a chthonic rite the offering is generally not consumed) passed and shared among the mortal participants in the rite.
Thanks
Here we thank the deities being honored, for their presence and for the blessings we have received from them.
Closing
Ending the ritual with a specific statement helps with making the move from sacred space back into everyday space.
This is, again, just a general outline and explanation; actual ritual scripts, when they follow, will be more detailed.
