I have a dedicated mp3 that I use when I work out. I know that most people just use their smartphones, but I worry about sweating on or otherwise damaging a sensitive and expensive piece of electronic equipment. Also, a dedicated mp3 player is optimized for its purpose and, I believe, delivers better performance. For the record I use the SanDisk Clip Jam with an sd card for added storage.
The only problem I have encountered is that many podcasts are only available through specialized streaming services, such as SoundCloud. Sometimes you can solve this problem by viewing the page source and finding the url for the mp3 file, but not with SoundCloud.
Thus, I was very please to discover this service, the SoundCloud downloader.
Here are some more thoughts about mp3 players.