Mendeley does this quite well. The options for folder organization are any combination of author/title/journal/year, in any order you want. The options for file names may also include any of author/title/journal/year, in any order, where you can separate by underscore, hyphen, comma or period.
One thing which could be improved is that Mendeley will not let you change the format of individual fields. For example, for author names, if there are up to 3 authors it will have all author names (e.g. "Johnson, Cohen, Livingstone_Nature_2013.pdf") but if there are more it will write it as "Johnson et al._Nature_2013.pdf". Also as you see it uses spaces and commas. So if the exact formatting is very important to you this may be a problem.
Note though that Mendeley has a special site where you can ask for new features and follow up on their progress. It works pretty well, though the timelines can vary (took them ages to get an android app).
As BrianDHall mentions, there are many other better reasons to use Mendeley - I definitely recommend it.