What is the easiest method to clean a bathroom floor that doesn't have a drain?
Since there is no drain and you want to take the lazy way out, I'd suggest using a steam mop similar to the one pictured bellow. We picked up one of these a couple of weeks ago and are blown away by how well it cleans.  

You just need to add water to the reservoir and then set the desired temperature, wait 30 seconds and you're set; barely any scrubbing required. Let the heat from the steam do the work.

As for water that you need to soak up after a bath or shower, a good old fashioned bath mat will do just fine.

Google Images of a steam mop

http://tsq.sewing.netdna-cdn.com/wp-content/uploads/Bissell-Select-Steam-Mop-94E9T-Head-Height.jpg