Iphone 4s using the Apple built in Podcast app. I have the app set to download content only when I'm connected to wifi and never use cell to save data costs, but that means if I'm at home, it automatically downloads in the backgrounds so I don't have to remember to do anything. And then later when I go to listen, if there's an episode I don't want to listen to, it's easy to just "mark as played" and let the phone take care of deleting it later. I use the "Up Next" feature to build lists frequently so I don't have to fumble around while driving to manually start another podcast when the current one finishes.
I don't sync to Itunes. If there's an episode of any podcast (I listen to several) that I want to save, I download it manually to my PC via gPodder.
I just got the Iphone last Christmas when I updated the rest of my family to Iphone 5 and took my daughter's old 4s for me. That's the first smartphone I've ever owned (and I've been working on computers since the 70's). I have to admit once you turn to the darkside it's easy and convenient. Prior to getting this phone, I had a LG enV Touch media phone that I synced to the PC where I downloaded the podcasts via gPodder and just played them as MP3 files but it was more cumbersome to deal with.