We Pagans love our incense! That has been the case for a very, very long time. As you might expect, the Minoans also enjoyed incense, and plenty of it.
The fresco above is from the West House in Akrotiri. It shows a young girl adding what looks like saffron threads to an incense burner. We know she's a girl because of her unnaturally white skin. And we know she's not an adult because of her hair - it's shaved, with a few long locks, an indication of a person who has not yet come of age.
The incense burner she's holding is similar to this one from Knossos. The image is from Sir Arthur Evans' Palace of Minos series, his chronicle of his excavations at Knossos. Evans found several incense burners of this type at Knossos - and he thought they were all flower vases! I suppose that's a reasonable guess, given that the fresco up top hadn't been discovered yet.
I can't imagine that the way the girl in the fresco is holding the incense burner is accurate - she would burn her fingers, since there are hot coals to heat the incense. Why she's not holding it by the handle, I don't know. Maybe the artist thought it looked more elegant this way?
This isn't the only kind of incense burner the Minoans used. They also used two-part burners with perforated tops, like this one from Ierapetra:
This is a two-piece vessel, with the base being shaped like a coffee mug with a large looped handle. You put the coals in the "mug" then add your incense, then slip the perforated lid down over it.
Here's another one of this type, from Amnisos, the port city that served Knossos. It's displayed beside a pyxis (jewelry box) and jewelry from Pachymmos:
Here's an article involving some experimentation with different kinds of Minoan incense burner styles. The section at the end of the article, with the two-level burner, shows that you can't make assumptions about how items were used. Experimental archaeology is so important, because often, the only way to truly understand is to do it yourself.
What did the Minoans burn in all these fancy containers? Here are some substances they had access to and that they likely used as incense:
Bay leaf
Cedar wood
Copal
Cypress wood
Dittany
Fennel seed
Frankincense
Gum mastic
Juniper berries
Labdanum
Myrrh
Oregano
Pine/fir needles
Pine/fir resin
Rosemary
Saffron
Sage
Terebinth
They may have added larger pieces (resin tears, small chunks of wood, whole seeds or berries or pine needles) to the coals. Or they may have ground up the ingredients to make mixed incense.
In addition to the temples, cave shrines, and peak sanctuaries, the Minoans had altars and shrines in their homes. They probably burned incense in all these places.
In Ariadne's Tribe, we use incense as an offering rather than for purification, because this is what the deities have led us to do in our modern spiritual practice. But we don't honestly know how the Minoans used incense. However they used it, I bet it smelled marvelous!