For purposes of this discussion, your river may be real or imagined, geographic or metaphoric.
I’ll start things off with the Guadalupe River. I drive or ride over it on my bike almost every day. It’s a rather sad excuse for a river, but it’s getting better. Thanks to clean up efforts the wildlife is slowing coming back; there have even been reports of beavers, which is pretty cool.
The Willamette River in Portland, yesterday. Prior to that, the Columbia River. It would be an interesting thread to see which rivers (the watery kind) people have crossed in their lives. My list would be quite long.
Does halfway count? If so the last real river was…
… when I went out to Belle Isle.
White River, which runs through the middle of Indianapolis, within the last few days. When I was attending school, I’d cross it at least twice a day, more if I had to make more than one trip. It ran right next to campus, and because of one-way roads and the parking areas I usually chose, leaving meant crossing the river, hanging a left, and immediately crossing it again.
The Podunk.
The last river I crossed was the Thames. I went under one way, and over the other.
Taking my partner to London City airport very early yesterday morning I took the Rotherhithe Tunnel one way, and for a bit of variety I came back over Tower Bridge.
It would be great one day to be at the front of the queue when they halt the traffic and raise the bascules although the chances of that happening must be pretty minimal. I’ve lived in London most of my life and I’ve actually only seen them raise the bridge on TV 
I cross the Hudson River twice a day for work.
I cross both the Potomac and the Rappahannock on a fairly regular basis. Not sure which was last.
I cross it often. And I always think of the B-52s when I do.
The Chao Phraya River, which cuts through Bangkok.
I cross the Mississippi fairly regularly since I’m in TN and my parents live in AR.
The Chattahoochie, but never over the same bridges from which Wayne Williams dumped his victims.
I drive over the Santa Ana River almost daily.
Been there, done that, got a video! I actually got stopped 4 or 5 times on Tower Bridge when they kept raising and lowering the bascules during the Olympics - got rather sick of it, to be honest.
This website might be helpful… Bridge Lift Times | Tower Bridge
My last literal river was also the Thames, I walked across Tower Bridge on my last day at work, which was also, by a nice coincidence, a metaphorical river that I crossed. Quitting was definitely the right choice, but it took me ages to realise that.
The Sheaf, after which Sheffield is named. I live only a couple of hundred yards from it so I cross pretty often.
I spent last week on the North Shore of Lake Superior crossing and (un)crossing many rivers. Baptism, Temperance (which has a huge liquor store upon it, ha) Tettegouche, Gooseberry, Knife and French among them.
They were all ice cold. Because of the iron rich soil and the many recent rains the falls along the rivers look like root beer.
On the way home I crossed the Mississippi and ultimately the Blue Earth River. They are just plain muddy.
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The Merrimack, several times a week.
The James. I was giving my family a tour of the city, and I wanted to show them how mighty and impressive the river is.
Crossed the Clyde (in Glasgow) by train yesterday.
Other rivers I have crossed (I feel a biography coming on) would include the Thames, the Forth, the Severn, the Hudson and the Susquehanna. If I ever crossed the Mississippi I must have missed it.