The greenest way would be to compost it. Seeing that you live in an apartment that isn't an option. I'd just continue to use plastic bags. Yes, plastic bags don't decompose but here is why I would continue to use plastic bags. Although plastic bags are recycable, usually only plastics #1 & #2 plastics are recycled. It's hard to find buyers for #3-#7 plastics, so most counties put these plastics into a landfill. Believe it or not the production of paper bags uses twice as much oil as the production of plastic bags. If you want to spend the extra money, you could use bio degradable bags.
you can get bio degradable bags that look and feel like plastic and some are scented we got them for nappy disposal
Get them at pet stores or search the web
Use zip-lock bags they don't leak the "stuff" out into the environment.
bury it under a few inches of dirt.
Most people use plastic bags. We usually do. But we're trying really hard to not accept grocery bags any more.
We live in an apartment that allows us to tie our dog out so long as we clean up after her. Trying to think of a way to clean up the poop, and dispose of it daily to keep it from smelling bad. Don't want to use any plastic bags. Any ideas?