There is no rock available in the Amazon, making it very difficult for the cultures living there to leave a lasting legacy, we therefor know very little about them.

The Peruan civilization that give rise to the Inka, did form around a series of rivers. Many small rivers running in parallel rather than one large though.

Mississipi was a cultural power center for north America. It was for example the home of the city of Cahokia, and probably the origin of the Algonkian shamanistic religion.

(All this is from 1491)

I'm here early so I assume some better historians after me will come in with more info, but one of the first things they'll point out is that Cahokia developed on the banks of the Mississippi.

With Monk's Mound being larger (at the base) than the great pyramid, and the city's population rivaling London at the time, I would argue that they left a very recognizable imprint.

Secondly, when it comes to Earthen structures like those at Cahokia, they degrade much more quickly (geologically speaking). Effigy mounds, found throughout the great lakes region and Mississippi valley, are incredibly plentiful, even after most of that land has been plowed for farming.

Add to that destruction the fact that most of those mounds are hundreds of years old (meaning the soil compacted making them less mound-y), and its easy to see why we might miss them. Some estimates say the dirt mounds have compacted to at least half their size, if not smaller (source: I worked at a mound site for some time, sorry I can't link). Still though, we find eagle effigies - mounds of dirt - that have a 1300' wingspan.. That's another sizeable imprint.

The last point I would make is that when you talk about the other cultures that have grown around big river systems, the examples you use are cultures in Egypt, and cultures in the Middle East.

Deserts

Water matters greatly to all cultures, but in places like the central and northeastern US, the landscape is braided with streams and tributaries. A small stream might not seem like much but it can support a community of a few hundred easily. Hence, those rivers are not reflected in the archeological record as central to the cultural development - though they were crucial for travel and the exchange of goods and information.

Do you mean to specifically exclude North America outside of the Mississippi from your question? Because much of Canada's population growth occurred along the St. Lawrence Seaway. As for native Americans, the Mississippi and many of the other major rivers in North America were important trade routes. As to why they weren't settled in the same way by Europeans (there are major cities along the Mississippi though not on the scale of say, Cairo) this has to do with the way America was colonized. The coast was a significant place to settle from a trade perspective with the old world, and in a time where land travel was slow and space in the colonies was still plentiful, there wasn't much of a need to move in land en masse to the Mississippi River.

I can't provide insight on why Central or South America were settled the way they were, but I hope this answers part of your question.

The shaft-tomb culture and later Teuchitlan tradition were centered around the highland lakes of Jalisco and we assume they made extensive use of the aquatic resources.

Teotihuacan is situated near the San Juan River and the city actually diverted it to run through the city to help clear waste. There are a series of drainage ditches in apartment complexes that run into the larger ones in the streets. The consensus was that the river water from San Juan was used to flush that waste away. Interestingly enough the ditches silted up before the fire in the Ciudadella and eventual abandonment of the city which may have been a sign of a loss of power.

Maya cities are built either near rivers like Palenque or near freshwater cenotes like Chichen Itza.

I remember discussing Monte Alban as an undergraduate because the city sits atop a plateau/hill with no spring. That means if the place were habitually occupied the rulers had enough power to coerce the commoners to haul up jars of water to them on a daily basis, which is no easy feat if you've ever driven up there for a visit and know how steep it is.

If you look at any major (or even minor) settlement in Mesoamerica you will find them situated near a perennial water source. People can't live with water.

First of all, be careful not to assume that because things are one way in one part of the world, that is the default that everything must be explained against. If anything, the whole coming together into cities with tens of thousands of people living in a densely occupied space and farms spread out along the river is a very strange thing, when you consider that small groups of mobile hunter-gatherers was the exclusive form until about 5000-8000 years ago.

But water is necessary, of course, and rivers everywhere are hugely important. Rivers allowed irrigation and population growth, accumulation of resources, founding of cities, etc. in the Andes. But the coast played a massive role too. The Pacific coast off Peru is one of the richest fisheries in the world and was a major source of food for people living along the coast, which is all desert save for about 30 rivers that run down from the mountains to the coast. Most of these rivers are little more than streams and aren't navigable, but the coast of course is, and people probably moved up and down it. The fresh water in the rivers allowed for agriculture which allowed for more intensive coastal fishing (growing cotton for nets and gourds to float the nets), and also allowed for irrigation to expand the river valleys and grow more corn and other crops. And this is where the earliest states and cities probably developed, with each river valley being like an oasis in the desert. The city of Moche, one of the largest in South America at the time, was built immediately adjacent to the Moche River, for instance. Other cities were farther from their rivers, but fed by irrigation canals.

But major societies and states also developed in the highlands, based around rivers. During the period when states first developed along the coast (the Early Intermediate Period, ca. 400 B.C. - A.D. 800) societies were fairly regional (there was some long-distance trade, but societies developed somewhat closed off from the others, based in their own region). The Santa River is one of the major coastal valleys but is also the only one that really draws from the highlands, running through the Callejon de Huaylas, a canyon between the Cordillera Blanca (snow-capped mountain range, the highest in Peru) and the Cordillera Negra to the west. Here it is a major glacier-fed river (not navigable because the water moves fast). This is where the Recuay society developed along the uplands on either side of the river. Other basins to the north, Huamachuco and Cajamarca, had their own societies develop along other major river systems, but those eventually feed into the Marañon River to the east (a huge river that is a tributary of the Amazon), whereas the Santa River flows to the Pacific.

Even the Inca, who first developed in the Cuzco basin for a couple centuries before they began their empire-building quest, developed along the Urubamba River--the Sacred Valley--where they located all their winter palaces and temples going downstream from Cuzco, including Pisac, Ollantaytambo, and Machu Picchu. Around A.D. 1450 they began to expand and conquer their neighbours. One of these was the Chachapoyas to the north, whose territory included the Marañon, the biggest river in highland Peru.

But rivers aren't always key. Tiwanaku, located in the high, cold, arid plains of Bolivia just south of Lake Titicaca, wasn't associated with any major river system, though they still needed rivers to feed their ingenious sunken garden style of irrigation.

So, basically, rivers are key, in the Andes too. So is aridity. In fact, aridity (either hyper- or semi-arid environments) seems to be a key component of everywhere where the first states developed. In part this is because materials preserve much better in arid environments so we have more detail about how these societies operated than those in more tropical or temperate areas, but aridity itself does still seem to be a key component for why people organized the way they did into cities and states, though these ideas eventually spread far beyond arid places, of course.