Podcasts are designed to be downloaded and played off-line. So playing them for class is not a problem, legally.
There are audio podcasts as well as video podcasts. The main thing you need to consider is whether the students will be interested and whether the podcast content will help the students reach their learning outcomes. For example, they might pay less attention to an audio-on ly podcast, compared to a video podcast (sometimes called a Vodcast).
There are SO many podcasts available that it is impossible to generalize about whether they are suitable or not.
I agree with Michael, podcasts are free and legal to download. When it comes to appropriateness, it depends on who is listening to what. But I like podcasts and find most of them highly useful. The idea is to help your students improve themselves by being exposed to English and possibly some content if you are intested in teaching them a particular subject. I recommend using ted videos and Six-minute English, Voice of America. These mostly work, yet the proficiency level of your students is very important.
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