This is the answer for the most part. They check to see if there is large bulk pickup on recycling days and will then, allegedly, pick it up by the weekend for free. When I moved out of my last apartment I just threw a ginormous pile of acceptable trash in my alley (where the cans are) and it was gone eventually.

I have an obnoxious amount yardwork that I need to be picked up, it's just chilling in my backyard until next Wednesday. You can also call and have it picked up whenever you want for a $100 or so fee.