First, I wouldn't be doing my job as an anthropologist if I didn't clarify that we see civilizations develop everywhere, regardless of how large their communities were. With that in mind, I am going to assume you are referring to cultures with large, 'urban' communities and with some degree of hierarchy and a control exerted over a region.
We 100% do see that develop along the Mississippi River, as well as along the Ohio River. There is a long tradition of human occupation of these regions, and probably the most identifying aspect of these cultures are the construction of earthern works, usually called mounds. Some of the earliest examples of this are found in the southern portion of the Mississippi River, this is where we see some of the earliest mounds being constructed, as well where we see some of the earliest evidence of larger communities developing. One thing you have to remember is that the Eastern Woodlands are resource rich, you have little incentive to form a community when it is far easier to just use the land around you; moving at various times of year as resources shift.
But back to moundbuilders, we see the earliest examples from around 5500 years ago at Watson Break in northern Louisiana. However, Watson Break isn't really a permanent community, it may have been more of a seasonal meeting place for regional groups. If you stay in that same part of the world around 4000 to 3000 years ago, we see the rise of Poverty Point, which in many ways is a continuation of Watson Break and shows developments we associate with later groups. Again, as with Watson Break, this site appears to have some ceremonial purpose to the people who built it; but also may have served as a settlement in some way. It is interesting as we see not only the construction of massive earthen rings, but also the use of plaza areas and the usage of posts in layout/construction. This is will be important later, keep it in mind.
There are lots of examples of early mound construction throughout Eastern North America, but in terms of urban development I am going to jump to the Woodland period, which kind of varies region by region but is roughly from 1000BC to possibly the time of European arrival. Again, this is varies a ton, so take these dates with a grain of salt. However, during the Woodland period we start seeing a lot more mound construction occuring in Eastern North America; almost like an explosion of ideas. We see Hopewell cultures develop in the Ohio River/Mississippi River regions, and with them we see more examples of ceremonial earthen works being built. Between 100BC and AD800, we also see major cultural changes in the introduction of maize and the bow/arrow. This is a gradual diffusion over time, but will be huge in the development of truly the first major city in Eastern North America.
Around 950AD along the Mississippi River near present day St. Louis, changes are occuring. A small community had existed in this area for a few hundred years, but suddenly things just changed. Archaeologically, we can see that this previous community was torn down and realigned along what can only be described as a planned community. We see a shift in how these people lived, but not due to conquest as there does not seem to be signs of forced assimilation. These changes would continue, culminating in the site we today call Cahokia.
This 1000-1050AD mark is called the Big Bang at Cahokia, and for good reason. We see rapid development of the city, thousands of people move to the area and construction begins on many monumental projects. Monks Mound, which was constructed by the inhabitants of Cahokia, a earthen mound that was 100 feet high and with a base roughly the same size of the Great Pyramid at Giza. It wasn't just a pile of dirt either, it shows signs of layering of soils and clays that would have helped it fight against erosion. It also seems to have been built by the local population without forced labor, these people bought into whatever idea fueled it's construction. We also see signs of astronomical alignment of mounds and structures at Cahokia, as well as the construction of a palisade that would have surrounded parts of the city. We also see the flattening of an area next to Monks Mound that we call the Grand Plaza, and this would have taken as much earth moving as the construction of Monks Mound. The area we associate with Cahokia would have extended roughly to the Mississippi River, and its influence was felt into Southern Wisconsin, Appalachia, the Great Plains and the South East. We can see this in the exchange of material both in and out of Cahokia, as well as the establishment of related settlements at places like Aztalan in southern Wisconsin.
Cahokia is estimated to have been home to 20-30,000 people at its peak, and it would take until 1803 for another city north of Mexico to actually surpass its size. Cahokia would maintain its dominance until around 1250-1300AD, but its decline was far from its end. We see that many who lived in Cahokia seemed to have moved to other communities and brought with them many of the idealologies they practiced at Cahokia. So it was less a decline and more a dispersal.
Fast forward to the arrival of Europeans, and we see groups in the Southeast who seem to still be practicing elements we associate with Cahokia and other related Mississippian cultures. So not only was Cahokia far reaching in its influence, but it also made an impact that was felt centuries later by people who did not even know Cahokia had once existed.
The Mississippi River has a rich cultural history that stretches back millenia, and can be seen reaching its 'height' through the development of Cahokia. Cahokia seems to fit the mold of what would be considered a city-state in the Old World, we even see possible indications of a ruling class who may have inherited their positions of power through a lineage. But it does not exist in a vacuum, it is the byproduct of cultural developments along the Mississippi that had been occuring for as long as humans had been in the Americas. It was part of a rich and vibrant past, and eventual future, that show all the hallmarks of any other 'cradle of civilization.'
Sources(Also further readings!):
https://mvac.uwlax.edu/PreEuropeanPeople/woodland_tradition.html
http://news.illinois.edu/news/14/0303cahokia_ThomasEmerson.html
Timothy R., Pauketat (2009). Cahokia : Ancient Americas Great City on the Mississippi.