While most of the ancient Greek festivals, rituals and devotional practices we hear about tend to be those held annually (or, sometimes, once ever few years), there are also traditional forms of devotion held on a monthly basis.
The Greek calendar was a combined lunar-solar one; see the HMEPA site for a reconstructed version of the Athenian ritual calendar. Some folks follow the HMEPA calendar, others the modern month while others still use the lunar cycle itself in these devotions.
The Noumenia was the first day of each month, which began with the new moon; it was considered a holy day, on which no public business was done or other religious observances scheduled. (Mikalson, “Noumenia” 291). Traditional offerings were relatively small, of frankincense, honey, garlands or cakes; it is uncertain which deities received these offerings but it is likely that they varied from place to place, and perhaps among individuals as well (293). Some modern Hellenic polytheists honor Selene at this time.
The second day of each month was holy to Agathos Daimon; the third to Athena; the fourth to Aphrodite, Hermes and Herakles; the sixth to Artemis; the seventh to Apollo; and the eighth to Poseidon and Theseus his son. (Roberts, Sokrates Loc. 2511) In keeping with this, a number of festivals to these gods take place on these days; for example, the festival of Artemis Agrotera takes place on the sixth day of Boedromion (Parke 55), while the sixth of Elaphebolion is the date of the Elaphebolia honoring Artemis Elaphebolios (Deer-shooter) (125)
Hekate’s deipnon or meal took place on the eve of the new moon, the last day of the month; at this time those who had the means would bring an offering of food to the crossroads for the goddess. (Parker, Miasma 30)
Mikalson, Jon D. “Notes and Observations: The Noumenia and Epimenia in Athens.” Harvard Theological Review. 65, 1972.
Parke, H.W. Festivals of the Athenians. Thames and Hudson: London, 1977.
Parker, Robert. Miasma: Pollution and Purification in Early Greek Religion. Clarendon Press: Oxford, 1985.
Roberts, J.W. City of Sokrates: An Introduction to Classical Athens 2nd ed. Routledge: New York, 1998. Taylor and Francis, 2001.