One partial solution to the problem would be introducing the style of public toilet you see in Amsterdam.[1] They amount to little more than a sewer grate with a privacy shield. There's no real need for cleaning these other than how you'd clean the street already, and it's a better alternative than pissing in a bush.

Of course it leaves the problem unresolved for women, and I'm unsure how you'd go about tackling that.

[1] http://1.bp.blogspot.com/-oNPlqgm_lXc/UZ7-2rB3arI/AAAAAAAANY...