There are lots of ways to find free Wi-Fi, but availability is often limited to a single area or restaurant. The Perth Cultural Centre free Wi-Fi network is a tad more extensive, extending over two large blocks in the city CBD.
The service, which is hosted by iiNet, is available in the area bounded by Roe, Beaufort, Francis and Williams Street (otherwise known as "the Perth Cultural Centre" or "that bit north of the railway station").
While I applaud the concept, it's not exactly a high-speed offering. When I gave it a quick run-through in the middle of the day using Speedtest.net, it offered barely tolerable download speeds of just 0.14Mbps, and uploads of 0.04Mbps. Bizarrely, the ping result of 8ms was much better, but with those speeds you won't be able to take advantage of it either.
Whether those results reflects overall performance or lots of people ignoring the "don't download large files stream video" rules is a big unknown. But at that level of performance, it's not really going to be much use if you want to do anything more than a quick Google search or two. Still, it is a backup option to remember if you do hit the Perth CBD and need a cheap connection in a hurry. (My previous testing suggests that the Next G coverage is pretty good if you're looking at paid options.)