I'd heard this for ages, but Tokyo in my experience had a lot of places to leave your trash. Not the giant trash cans I see stateside, but small bins next to many of the vending machines which are everywhere. The streets were pretty clear of litter though. The lack of litter was still impressive.
You have to have a recycling bin next to drink machines. These bins are only for bottles and cans, though. You shouldn't put trash in them. Convenience stores are required to have trash bins, so that's the easiest place to dispose of something if you need to.
I don't think this has anything to do with terrorism, though. There is a culture of take your trash home with you. Even at work I was not allowed to use the trash bin for anything except work trash. Trash from my lunch, etc, I had to take home.
It seems to be extremly impolite to put anything into the vending machine bins but the bottles you bought from that exact machine though. My Japanese friends were visibly uncomfortable at the idea. Same for any convenience store bins - only what you bought there is fine.