The city of Helsinki offers a free WLAN-service for residents and travelers alike. No passwords or registration required. Just look for "Helsinki City Open WLAN" from the available networks. The following map shows the outdoor hotspots around Helsinki.
NextMesh
[Photo from visithelsinki.fi]
There are plenty of commercial services that offer free WLAN around Helsinki as well. NextMesh covers most areas popular amongst tourists, from the city centre’s Kamppi to Kauppatori and all the way to the Kaivopuisto.