I don't worry too much about it. After about 5 days or so, I start to think that a shower might be nice, but prior to that point, it's not a big deal.
If I can get to a convenience store (or restaurant, or grocery store, or whatever) washroom, I'll have a "bath" in the sink using water and paper towels, and possibly the soap in the dispensers.
A lake, ocean, or stream is also useful ... I'll wade in (cycling clothes and all) and splash around for a while (no soap, of course). Sometimes, especially at ocean beaches, they have showers for people to wash off the sand, so I'll have a quick shower there (and I might use soap there).
As for cleaning things like dishes when water is scarce, I have scrubbed them out with sand and then given them a quick rinse whenever I got to a water source.