How do I connect components when using a circuit board with pads but no traces?
Here's another interesting method of perfboarding, this guy does it with SMD components and really tiny wires. He made a special little wiring pen for laying down the wires:

Progressive Wiring Techniques


(source: davr.org)  

Personally I make solder bridges for pins that are in adjacent holes, for things further away I use thin wires (though not as thin as the ones above) in adjacent holes to the things I want to connect.

Top of my board: 



Bottom of my board:



And no, I didn't just wire those pins all on the fly, I had drawn up a detailed schematic in advance. There's no way I could keep track of so many wires just in my head.