Rituals of Ancient Mesopotamia

The religion/s of ancient Mesopotamia (Sumer, Akad, Assyria, Babylonia, Ebla, and Chaldea) influenced the subsequent religions of Canaan, Aramaea, and ancient Greece, including Christianity, Islam, and Mandaeism, and, despite the fact that much has been lost over time, has historically the oldest body of recorded literature of any religious tradition (including the Enuma Elish, or Epic of Creation) sufficient to gain a good overall image of the mythology, beliefs and practices (including astrology) of these polytheistic people. 

In terms specifically of their cultic practices/rituals: each city was home to a deity and would have had a ziggurat/temple as the dwelling place of their patron. This rose in a stepped form to the sky - a staircase for the gods to ascend and descend, and sometimes included a holy tree in a holy grove, which was the central point for the rites performed by the King, in his role as 'master gardener'. The presence of the deity was symbolised by an image in a separate, special room, the idols being consecrated by nocturnal rituals, including washing, believed to give them life and to open their mouths so that they could both see and eat. The idols were dressed and served food twice a day (possibly shared by the King and/or priests), as well as receiving sacrificial offerings, and were sometimes also taken out on hunting expeditions. The Priests included sa'ilu (askers) and baru (observers), as well as mahhu (ecstatic seers). The deity was also served by singers, craftsmen, swordbearers, and master-diviners.It was believed that the deities expressed their will through words (amatu) and commandments (qibitu), which were made manifest in events and things. As well as the deity in the Temple, individuals also paid homage to personal deities in private devotions, and practiced diviniation, through observing omens and rituals such as casting lots. Both individuals and Temple priests used incantations (šiptu) to ward off demons. Amulets and exorcisms were also used. 

Mesopotamian rituals are known to have included incantations (for example, for the removal of snake-venom), petition/letter-prayers (please for relief from illness), Ikribus (used in divining and as benedictions), and Šigû (lamentations). Namburbi were performed during rituals, especiallyto counteract an evil fate indicated by an omen, while the Maqiû series of incantation was intended to counteract kišpū (black magic). The incantation series Šurpu, inscribed on cuneiform tablets, consists of a long confessional of sins, ritual offences, and unwitting offences against the moral/social order and breaches of taboos. It begins 'Enūma nēpešē ša šur-pu t[epp]ušu' ('when you perform the rituals for Burning... '. The burning referred to is of dough which had been applied to or wiped over the person, thus transferring the sins to the dough which would be burnt. Other items such as garlic, onion peel, or red wool, representing transgressions, could also be thrown into the fire, while the following incantation was recited:

My illness, my weariness, my guilt, my crime, my sin, my transgression,

The illness that is present in my body, my flesh (and) my veins,
Be peeled off like this garlic so that
The fire-god, the burner, consumes (it) today!

May the curse leave so that I may see the light!

Another cuneiform tablet related to items used during purification, including the words, 'Your hands are washed... you are holy; your hands are washed... you are pure.'

Refs:

https://en.wikipedia.org/wiki/Ancient_Mesopotamian_religion

https://www.britannica.com/topic/Mesopotamian-religion