Many places don't take cash, Yaki-Da[1] in Gothenburg for instance. Many (all?) restaurants on Tredje Långgatan as well, and I've also seen several places in Stockholm where they won't accept cash.

Also, many cities' communal services like busses and trains no longer deal in cash.

[1]: http://www.yaki-da.se (In the bottom right says "Yaki-Da är en helt kontantfri restaurang och nattklubb.")