there is some discussion about unit movement and rivers, depending on whether rivers are on tiles or not. Personally, as I say in my first post, I want to see rivers on tiles to emphasize the movement-enhancing properties, but this involves some complications too (for instance how to decide/show which riverbank the unit is on).
I think rivers on tiles would make for a nice change now that we've had two games with rivers between tiles, both game-play wise and aestethically. I agree it makes more sense for cities (at least large ones) to lie on the river rather than on one side.
If terrain with different altitudes is used - as some people on this forum seem to want - rivers on tiles makes absolutely more sense. Rivers between tiles of different altitude would just seem plain weird. Also, such a revised system would give much better river placement and courses when generating maps. A relatively simple algorhitm could be that a river always flows to the lowest adjacent tile (if several of same altitude we can choose or randomize). If a river runs into a depression it will fill up and create a lake before it runs to the ocean. Flood plains will occur in low flat areas. This would help get rid of many of the absolutely silly rivers we see in Civ4 (seriously, which river starts as a flood plain and then goes into the hills before ending in a small lake

).
Oh, and it might maybe make it fun to make dams and canals too (though I don't know exactly how)
I would love to see a more natural look to the game and to make rivers more useful. One thing I've noticed absent in CIV games is the lack of a scalar property in objects. Either it's there or it's not; no differing in size. (Hills ARE NOT small mountains. They're HILLS.) And most obvious of all is the lack of scale in rivers. There should really be different sizes of every geographical object (let's say three to simplify yet still be distinct). There should be Size 1 rivers, like the headwaters of all rivers, which are readily fordable, but too small to navigate with all but a wooden raft, Size 2 rivers, which are unfordable without a bridge and can be navigated with flat-bottomed boats, and then Size 3 rivers, which can accept ocean-going vessels with ease, and can only be bridged across with great difficulty or not at all. (ditto the above for man-made canals) Also, their size should be relevant to the map so that a player can tell just how immense the river is upon discovering it. (Maybe a player should even get naming rights, but that might be covered somewhere else. It'd still be fun though!) Accordingly, with navigable rivers will come river vessels that can only operate on such. And until the proper level of engineering is discovered, the larger rivers will be impassible except by boat. Foot traffic will have to be ferried across, restricting traffic (units).
If elevation is added to the game (even simpified), then there should be at least one distinct watershed (or two if there is a continental divide, ie. mountain chain) for each regional landmass that collects and drains all the water into a common basin. Think of the Mississippi watershed that covers thousands of square miles upstream and branches out to collect most of the water that falls east of the Rockies and west of the Appalachians. The Mississippi is enormously large at its delta, but the farther up you go it gets smaller, especially once it starts branching out into the Ohio and the Missouri. So a continent should have most if not all of its smaller rivers draining into something akin to the Mississippi - a Size 3 river let's say, if enough tributaries run into it. Realistically connecting rivers like this will enable all cities located on any part of the river system to trade with any other river towns. And historically speaking, trading on any scale at all was only possible originally by using a nearby river due to rough or impassible terrain, hostile natives, and the extreme length of time it took to travel on land. (Early roads should reflect this, and not be given a movement bonus, imo. Most terrain w/o roads should be impassible or very nearly so, at least hills and mountains. Elevation will hopefully correct this.) Even then most (smaller) rivers had rapids that were hard to navigate, or waterfalls, which stopped traffic completely. But with engineering, even these obstacles were overcome via locks and dams. A whole new approach to playing CIV could be opened up if we were allowed to survey our land, dam up a river to make it navigable upstream or to create a large inland lake, build canals between two bodies of water to enable navigation and trade, and a host of other terraforming ideas. For those of us who prefer peaceful pursuits to war and destruction, I would think that being able to improve one's homeland geography would be a great ability to have.
Here's a solution perhaps concerning the problem of rivers going through squares instead of around them and city sites on such squares. I imagine that most if not all towns probably did start on just one side of the river (who wants half the town on the other side?), and even a city's governing boundary today is almost always on just one side of any large river (if it looks built up on both sides, it's usually another smaller city, and quite possibly a different state, at least here in the US). Most cities were founded on one side or the other, because there was just no reliable way to ford the river, and I imagine the town's founding fathers didn't see permanently splitting up the town as a good thing. Yet cities were always settled next to a river if they had a choice because of the natural advantages of trade when they did start building boats. So l propose this: let the river generally follow between squares, meandering deep into them and even cutting off a large corner of one when bending to make it more appealing. Whichever side of the river a square's center point is on, then that whole square (albeit smaller) will be considered to be on that same side. And to account for any square's larger (or smaller) size that borders a meandering river, the hammer/food/coin value of that square needs to be adjusted reflect its gain (or loss) in land and material. It would certainly make surveying city sites next to rivers that much more interesting and challenging!!
Something else that needs to be enforced upon river cities is this: A city founded next to a river may not use the squares on the opposite side of the river just for that reason. (ditto for coastal cities) But to keep a city's potential from being hindered (and to keep hordes of people from slamming me for such a stupid idea), let any of a city's (a town's) nine squares count as far as being next to a river is concerned, and/or let a town's starting 3x3 block be allowed to shift up or down one square and left or right one square. In this way a city can be founded next to a river, not illegally expand to the opposite side, and still be a potentially powerful city, just with all of it's growth being restricted to one side. When a city's culture grows, it can expand across the river (it's just culture), but it may not use any squares on the other side for production (allow it to expand outward on its own side instead). Then it would actually be possible (and allowable) to have two cities adjacent to each other, just on opposite sides of the river. (they better get along)

There's plenty of examples of twin cities here in the US, as I'm sure there are all over the world, and some are powerful metropolitan dynamic duos! It always befuddled me how a city founded on one side of a narrow channel could claim and use land on the other side of perhaps a completely different continent! Natural geographic borders need to be enforced just the same as man-made political ones.