Many side streets have traffic lights that are triggered by the weight of a car on the road and are not triggered by the weight of a cyclist. In this situation, either wait for a car to arrive (this could be a long wait) or carefully look to make sure that the way is clear and then cycle across, in the same way that you would cross as a pedestrian.
If these traffic lights are on a route that you take regularly, write a letter or email to the council pointing out that this type of restrictive infrastructure is at odds with their commitments to increase the number of people using active transport, and to improve the perception of the safety of cycling.
The Road Code has some hints for triggering these lights (Under 'vehicle detectors'): http://www.nzta.govt.nz/resources/roadcode/cyclist-code/about-cycling/cycling-through-intersections.html
It's not perfect and I've only managed to trigger it only once using this technique.
It's not the weight of the vehicle, it's the amount of metal that's picked up by the wires in the road. Look for the giveaway pair of rectangles and ride on the line right up the middle – that's the most sensitive spot. Of course if you're on a trick carbon fibre bike you might be out of luck, but most steel-rimmed wheels will trigger the lights if the inductance loops are properly adjusted.
@Steve: A friend glued rare earth magnets under her frame to do just that. I think the older triggers don't pick up on those though
If you want a bit more info about the "science" behind cycle detectors, have a look at this article from CAN's ChainLinks magazine:
http://can.org.nz/resources/a-disturbance-in-the-force-traffic-signal-detection-of-cyclists
Practically speaking, the general gist is: ride ALONG the lines cut in the ground to have the best chance of being detected.
Thanks Glen, that's very helpful!