I'm a serious NPR junkie so this was an issue for me when I picked up my Droid. I have never had an iPhone, but do own an iPod Touch and am a big fan of the official NPR app.
On Android I've been able to stream my local NPR station's Shoutcast stream through the DroidLive Lite application. It's not as seamless as the iPhone app, but it's now the default application for any mp3 stream I download through the web browser or MyNPR.
When using MyNPR to find stations, be sure to click on the search results labeled "MP3". This should open the stream directly in DroidLIve. If you choose "STREAM" the browser will open and download a file that Android can't use.
There is a paid version of DroidLive, but the Lite version has been fine for my purposes.