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There isn't a quick, light free way of crossing that part of the city, unless you're willing to rlj all day and ride o many pavements.
If I were crossing Blackfriars to get to Shoreditch, I'd go up Old Bailey, around Smithfield, Golden Lane, central > lever street and then through Hoxton.
I haven't done this route regularly in a while, but when I did I went over Southwark, across queen street, up Coleman Street and join Moorgate. Then suck up some traffic lights and take a right on Tabernacle street towards Pitfield street.
I'm definitely using Blackfriars or Southwark, the issue I'm having is coming to a grinding halt or ending up on unpleasant roads when I get north of the river. Either Blackfriars ends up putting you on Clerkenwell Road, which is pretty busy or onto the multi lane roads running east near Barbican. Southwark Bridge dumps you into a bunch of traffic-light controlled junctions, until you get up north of Finsbury Circus. It's a quick way of doing that bit that I'm really after.