I'm using Visio 2013 and I'm not able so save my drawings with a transparent background (background always ends up with white color)
The sheet background is "no background".
When I click "save as .PNG" (or .gif) there opens up a options menue. There I can also choose a background color but there are just colors and no "transparent" options.
Thanks in advance!
In your PNG options simply tick Transparancy Color:
Don't amend the colour, leave both as they are, and save.
In this example Drawing1 was saved with this option ticked, and Drawing 2 without: