Well, I originally wanted to use solder bridges between the pads to make nice clean lines along the board, but for some reason I couldn't get the solder to bridge easily, I was making too much of a mess that way. Good job I tried it out on a bit of scrap perf first!
In the end I decided to connect all the cathodes in each horizontal row of LED sockets together using a single strand of stripped bell wire to give the solder something to adhere to. That's worked well enough, even if it's a bit ugly. But it's not as if that side will be visible in the final build anyway.
I've just finished that a few minutes ago. Now I've got to make lots of small bits of wire and thread them through at 90 degrees to the first set of wires and then make connections to all the anodes in each column of LEDs.
That's going to keep me busy for another day or three!
I'll upload more pics when it's done.