Downriver from London Bridge is Tower Bridge. The most recognisable and celebrated of Thames crossings was opened in 1894. It's a must-cycle experience. The busy traffic hems you in a bit on the bridge itself, with some unpleasant ridges right on your cycling line and railings by your side, but the speed limit is 20mph, which just about makes it enjoyable. The most dramatic approach is from the south, where the angle of the road combined with the bridge's series of vault-like arches gives the impression you're cycling into the oesophagus of some fabulous sea-monster.
Watching its roadway split and lift to allow river traffic to pass underneath, which it does around twenty times a week, is dramatic too. Lift times are on the Tower Bridge website; lifts are also catalogued on the bridge's own Twitter feed, like a perkily repetitive primary school teacher you get off with at a party to your swift regret.
Just to the north is the Bloody Tower, which gave the bridge its name. (The fact that the selected design happened to be two towers was coincidental.) In fact, if you're stuck in a car while the bridge is up, as Bill Clinton's motorcade famously was in 1997, you'll probably call it Bloody Tower Bridge.
The view from the walkways at the top of Tower Bridge is described as 'stunning' on the bridge website, though that also describe the bridge's forthcoming maintenance works programme as 'exciting'.
You can tell locals from tourists because locals would never pay the extortionate entrance fee to visit the Tower or to go along the walkways at the top of Tower Bridge. You can tell Yorkshire-born incomers from locals because they will find a way to do them for free. And then won't tell you what it is.
Tower Bridge is undergoing major maintenance work over 2009-2010 and will be closed for several periods. There's talk of a ferry replacement but don't count on it. Your only alternative if it's closed is the hike back to London Bridge.
From here along the north side is a complicated but very enjoyable and all traffic-free amble through docks and alongside ornamental canal, including Britain's bendiest bike path. Along the south side is another complicated but largely traffic-free run, mostly along the picturesque and chain-restaurant riverfront, and at one point apparently going through someone's back door. I've knocked up a Google map of both routes between Tower Bridge and Greenwich (to get to Rotherhithe Tunnel, come off the marked route at Elephant Lane). Either way, it's a mile and a bit to Rotherhithe Tunnel.