A couple things: your URL only gives the podcast for Wednesday, April 8, not every day. You'd want to subscribe to the feed (I guess you're forced to use iTunes) and then get the properties of the podcast from iTunes and type it into your Chumby, or talk to the nyt folks to get the pls from them to type in...
...or you can use the 2.6.83 Control Panel beta, which has support for the New York Times podcasts, including using them for alarms. I concur with Duane!
The other thing: Under My Streams I created a new stream of type MP3 and entered your URL and it worked for me. I can only guess you had a typo -- the url you gave would be easy to mistype.
wayn3w