What's the most effortless way for "cloud" back up of photos taken with an Android phone? After I take a picture with my phone, I want it to be synced (and/or shared) to the "cloud" shortly thereafter. (Say after a few seconds when I have wifi connection but not necessarily right away when on 3G). Does such an effortless synchronization system exist? Can a Picasa or DropBox account already do this but I haven't taken full advantage of it? Am I missing something?

I can't seem to find an official Picasa app by google. This Picasa/Android question I found seems to tell me it doesn't exist, although it's an old question. 

The option to auto-share photos backed up this way would be worth mentioning. Non-picasa solutions are also well worth mentioning.
If you sign up to Google+ you can do this with the Google+ app now. 

There is an Instant Upload option in the app. This syncs any photos (and videos if you set the option) to an Instant Upload album in Picasa.