But can one do this in a polytheistic context? ADF's COR includes the step of asking a gatekeeper deity to open the gate between the worlds, so that the gods may hear us and interact with us. If I think or say a prayer without doing that, is it likely to be heard?
So I e-mailed Ceisiwr Serith, author of A Book of Pagan Prayerwith that question. Fortunately ADF's wise old elders are eminently accessible and helpful, and he responded just a few hours later:
Sure. I do all the time. In fact, my morning and evening prayers aren't in the Core Order of Ritual style at all. (The morning prayers are here: http://ceisiwrserith.com/ritual/practice/morning.htm). I often pray to the Vedic goddess Vac ("Word") before writing, to Cernunnos when I need calming, to Ganesha and Isis when flying, and so on. Whether made up on the spot or previously-memorized,, the sand-alone prayer is a powerful thing.
That said, in groups more elaborate rituals are good things. There are a number of reasons for this. It's good for a group to be on the same page; it increases the power of the ritual. A set pattern is good for this -- it allows everyone to know what comes next, so they can stay together, it allows the prayers to be additive, rather than muddled, and thereby increases the order, and thus the beauty, of the ritual (and beauty is at the center of the IE cosmos). In order to be more elaborate, there needs to be a structure, and the COoR provides that. The COoR is devised to fill in the necessary holes, and to be IE in spirit, thereby connecting us to those who worshiped these deities in the past. They must have been on to something, and these are the sorts of rituals the Holy Ones are used to, and presumably like.
Most important, the rituals express both the ADF/IE cosmos and the way we relate to our deities. They teach us these things and reinforce in us, to the point where it becomes easier to relate to them, and to make those spontaneous prayers. So participating in the more planned out rituals actually strengthens the non-planned out ones.
If possible, then, it is good to have structure to your own prayers. This will set up echoes between them and the public ones, as well as with the cosmology and ritual theory behind ADF practice.
At the minimum, it is best if you can do your prayers in the presence of fire and water. That's standard IE practice, for which there are a number of reasons. Fire can be seen as Order, and water as Chaos; both fire and water are purifying; water is the border and fire the center; and so on. If you can, use the actual objects, but there's a fair amount of discussion off and on on the Dedicants' list as to what to do when you can't, especially when you can't use fire -- electric tea lights, pictures, stones, etc.
It's worth making the effort to do this. But if you can't, because of where and when you are, then go ahead and pray. In a sense, such prayer is less than ones in the presence [of fire] and water, and in a sense they are exactly the same. Just do your best. I'm not saying the classic Neo-Pagan/Neo-Wiccan platitude of "do what seems right," but "do what is best (under the circumstances.)
I recommend having an arsenal of set prayers that you can whip out at a moment's notice -- ones for calm, for when you are afraid or sick, ones when you want to praise, thank, or feel connected with a deity, ones before eating or drinking, and so on. The more you have of these -- and they can be just a single sentence (http://ceisiwrserith.com/ritual/theory/piety.htm) -- the more you will be putting yourself in tune with the divine and the cosmos.
Above all, never let the fact that you don't have the time or equipment to do an elaborate ritual stop you from praying.
(One more thing. Using objects in prayers, and not just words, makes sure we remember that it is not only thoughts and speech that can be sacred, but the material.)