Downriver from Chiswick Bridge you cycle a few hundred metres along a lane and then riverside path. Go just under the railway crossing of Barnes Bridge and hoik your bike up the two flights of steps to platform level in Barnes Bridge station.
You can wheel your bike across the footpath. The structure was built in 1895 and is Grade II listed.
From the bridge you'll see a blue plaque on the wall of a handsome riverfront house with a latticework balcony (right): it was the home of Planets Suite composer Gustav Holst (1874-1934), who was a keen cyclist (and very English, despite the name). Once he had to deliver a score to a colleague on the south coast, and decided to cycle there. When he turned up at the front door, the wife of his colleague assumed he must be a mere delivery-boy, rather than the celebrated composer of the Planets. We all know the feeling.
She sniffily directed him to the tradesman's entrance round the back. Unfazed, Holst went round the back to meet his colleague with a cheery 'delivery for you, sir'. A lot of Holst's music is appealing and modern-sounding thanks to its odd metres, robust rhythms and haunting melodies: try the Perfect Fool suite, and the Hardyesque bleakness of Egdon Heath.
Down the steps on the other side, turn right and cycle along the riverside downriver past sports grounds and boathouses. There's one little set of steps to lift your bike over but the rest is cyclable.
There are refurbishment works going on from now until 'early 2010' which will involve the bridge being closed for a few weeks at a time at various periods, so don't bank on it being crossable for a while. If the bridge is closed, or if you don't fancy coping with a lot of steps, then cut out both this bridge and the similar Fulham Railway Bridge a bit downriver.
Between here and central London's South Bank, it doesn't make much difference which side of the river you're on – there's not always a 'best side', and you'll often be nosing your way round streets away from the riverside. So if you have to miss out a bridge, it doesn't matter. The trick you can pull is that later on, when you cross the Jubilee Bridges, you can count them as either one or two crossings (as they're two separate bridges but feel like one). Choose whichever option ensures that you're on the South Bank after the Jubilee Bridges, or it'll all go wrong at Woolwich.
Whichever side you're on downriver from Barnes Bridge, continue along the riverside path up to Hammersmith Bridge.