This is a much more complex question than it seems.

The concept and intellectual understanding of the underworld and the afterlife in ancient Iraq was only just formulating during the Early Dynastic Period to the Third Dynasty of Ur in the third millennium BCE, and for this reason different sources have different, and conflicting, viewpoints on these topics.

For example, various Sumerian literary, administrative/archival, lexical, dedicatory, and magical sources give varied details about precisely where, in what direction, and how one could travel to the underworld ('under the ground, far away to the West, above or below the /Absu/ (the 'subterranean ocean'), at the base or at the top of the eastern mountains’, or across the 'River of Death', and so on).

It's important to note that we have a paucity of Sumerian sources on these issues through to the Ur III period. Sumerian was still being used administratively in the Old Babylonian period and texts were being copied into the first millennium, but as time progressed it's clear that the precise meaning of portions of lexical texts and aspects of the language become increasingly obscure to the contemporary scribes. This results in identifiable misunderstandings in translation from Sumerian to Akkadian, and the obscurity of the language and outstanding issues in Sumerian grammar further complicate our understanding of what third millennium Sumerians believed. The situation is therefore somewhat more complex than is conveyed by most secondary sources (even for specialists), and/or they tend to base themselves on the later, clearer, and much more developed Akkadian literature -- which in many cases constitutes later-period rationalization and literary evolution from the Sumerian originals.

An example of this are the details present in the Akkadian "Descent of Ishtar to the Underworld", within which we learn that ‘travel to the underworld 'is one-way, that it is devoid of light, and that "dust is their [the dead’s] food, clay is their bread". These details are not conveyed in the Sumerian original, the Descent of Inanna. However, the notion that there is nothing edible in the underworld does have its origin in Sumerian literary texts, such as the "Death of Dumuzi":

   

To the Eshlam, to Ereshkigal, [Dumuzi] was trapped [in the Underworld]
Food is there but cannot be eaten, since he was trapped [in the Underworld]
Water is there but cannot be drunk, since he was trapped [in the Underworld]

   

The notion here is that while options exist, because Dumuzi 'was trapped' in the underworld, the food and water don’t provide sustenance — a commonly-known issue that was the case for all other residents.

Some other specific details from the Sumerian period are known. There is a clear belief that when a person died, their spirit travelled to the underworld. Dead family members were sometimes buried under the houses, and remained an important part of the family unit. To sustain their spirits in the underworld, offerings were made in the interments and in ritual thereafter. Funerary texts outline that living family members provided bread, water, and ointment offerings to the ancestors, represented in the houses by statues of individuals, which were also ritually clothed. Having many children and descendants was important, in part to conduct these ritual acts, which then provided these 'necessities of life for the dead' in the underworld. Some texts even suggest that the living conditions and even social hierarchy of the spirits in the underworld depend in part on how many descendants and of what type and number they had.

Sumerian magical texts suggest that these ritual acts of providing offerings to the dead prevent spirits from becoming hungry and angry, which could then result in their rising up in the world of the living as /galla/ (a complex concept and term that initially means 'a spirit resident of the underworld'. However, this concept evolved over time, and eventually comes to describe a 'vengeful spirit' that can commit evil deeds against the living, up to and including murder. Incantation texts exist in Sumerian to protect oneself against these vengeful forms of /galla/, which some secondary literature refer to as a type of 'demon' (a bit of an inappropriate anachronism of sorts).

So, in context, the Sumerians believed that it was /galla/ that were "miserable, hungry, and thirty in the afterlife" — but that this was preventable for one’s ancestors and dead loved ones by providing regular offerings. Providing offerings was the ongoing duty of a good descendent, as spirits that did receive offerings still lived in the "dark, dusty" underworld, but at least the living could mitigate some of the suffering.

It is important to differentiate third millennium Sumerian thought from later Sumerian and Akkadian texts, but this is much more difficult than it seems and tracing specific ideas can be challenging (this is what motivated me to respond!) However, this depiction and understanding of the underworld had significant influence on later literary traditions and conceptions of the underworld and the afterlife across the ancient world.

   

Sources:

 

One of the best monographic treatments of the matter with primary source in context is:
Katz, Dina. "The Image of the Netherworld in the Sumerian Sources". CDL Press, 2003.

 

Overview history and archaeology by prominent scholars:
Kuhrt, Amélie. "The Ancient Near East: c.3000–330 BC". (2 volumes). Routledge, 1997.
Van De Mieroop, Marc. "A History of the Ancient Near East, ca. 3000-323 BC". Blackwell, 2015.

 

Primary source in translation by prominent scholars:
Dalley, Stephanie. "Myths from Mesopotamia: Creation, The Flood, Gilgamesh, and Others". Oxford, 2009.
Black, Jeremy. “The Literature of Ancient Sumer”. Oxford, 2005.