Ancient Sumerians saw their purpose on the Earth as being to serve the gods. It was believed that the gods had instituted human society as a reflection of their own administration of the Universe, over which mankind has been given dominion on the understanding that humans are to be caretakers of the planet and all its creatures. (This idea would eventually make its way into the Abrahamic faiths via the Semitic Canaanites; see Genesis 1:28).
Mesopotamian society was therefore arranged around the worship of the gods. Society on Earth was supposed to reflect the divine order, and the ideal function of human society was to strive towards such an order. It was believed that the gods dwelt in their statues at the great temples, and these statues were cleaned, fed, and attended by temple personnel. Obviously, modern society is not arranged in such a way, so how should this service be conducted in the modern day?
There is no equivalent of the Ten Commandments; no list of rules that are meant to be followed. However, we should strive to uphold order and justice to the intentions of the gods. We can discover what this means by looking at actual cuneiform tablets. The following appears in the opening of the code of law of the king Ur-Nammu:
"Then did Ur-Nammu, the mighty warrior [...] in accordance with the true word of Utu, establish equity in the land, and banish ill will, violence, and strife."
Furthermore, while the king was custodian of the Earth and its people for the benefit of their gods, he did not have free will to act as a tyrant. Grave warnings can be found in Mesopotamian religious texts against the mismanagement of the Earth by the kings. A tablet commonly known as "Advice to a Prince" contains the passage:
"If [the king] mobilises the people of Sippar, Nippur, and Babylon to forced labour, Marduk, the wise, the majestic, the noble, will turn his land over to the enemy. His troops will be mobilised by the enemy to forced labour. For the great gods, Anu, Enlil, and Enki, who dwell in the heavens, have decreed the freedom of the people from such obligations."
Thus we see that the ideal social order that the gods intended was not a tyrannical one, but one in which fairness and equity prevailed; justice should be administered on a proportionate basis, and there were serious cosmic penalties for rulers who overextended their authority. Therefore, in the modern day, one way to serve the gods is to uphold ideals of justice and fairness, and of a well ordered society, and indeed to stand up to injustice and tyranny.
Besides the social aspect, a great focus of the worship and service of the gods is distinctly personal. Although there are no great temples to the Mesopotamian deities any more, it was still a common practice to maintain an altar in one's home in Mesopotamia, and this practice should be carried out in the present day if possible.
It is neither required nor expected to worship all of the thousands of deities of the Mesopotamian pantheon, nor even all of the "great gods" of the pantheon. Personal, local, and civic deities played a major role in Mesopotamian society, and if a person feels called by, or feels a connection to, a particular deity or even a small number, then it is perfectly acceptable to focus worship on such deities.
Factors that are of great importance to the Mesopotamian gods are sincerity and sacrifice. We, as individuals, do not have the capacity to build lavish ziggurats or statues of lapis lazuli, nor is it expected. However, what is expected is that worship is offered to the best of our abilities and resources.
A person may be compelled by circumstance to worship in secret, and if this is the case, it is considered acceptable to offer such private prayers and offerings as can reasonably be accommodated. For those who are able to practice openly, forms of worship may include:
Unlike some paths which place a lot of importance on the formula of the words spoken, Mesopotamian faith focuses more on the spirit and the intent of our actions. We are not required to follow specific formulas, nor to recite particular prayers, but we are to offer as much as we are able. This is meant literally - the obligation is to do what we can in service of the gods, but also not to demand more of ourselves than we are capable of giving.
Owing to reasons of health, finances, energy, practicality, or many other factors, it's not always possible to dedicate a lot of time, space, and effort, and we accept that as a natural part of life. A sincere but modest effort will be appreciated much more by the gods than one that is ostentatious but made for show.