EDIT: Craig beat me. And how. Guess I took the long road. Browser location bar? Pshaw! 
That URL loaded fine in my mp3 player (foobar2000).
Your problem could be that your MP3 player doesn't follow HTTP redirects.
For example, when you visit
http://www.lightningmp3.com/live/file.php?fid=3243
The server issues a HTTP 302 (redirect) pointing to
http://www.lightningmp3.com/live/35595_keepintouch.mp3
If you add
http://www.lightningmp3.com/live/35595_keepintouch.mp3 to your playlist, that might work for you.
If that doesn't work, it could be that they're doing something to block your MP3 player's user agent or something. If you're feeling really curious you could always use a packet sniffer like Ethereal to troubleshoot from there.
The server seems to respond fine when no user agent is specified, so I'm guessing your player doesn't handle redirects.
If you're feeling especially geeky, try using your favorite Telnet application to connct to
http://www.lightningmp3.com on port 80 and type the following:
GET /live/file.php?fid=3243 HTTP/1.1
Host:
http://www.lightningmp3.com
(hit Enter twice after the ".com" to issue the request). Congratulations, you're now a crude HTTP client. You'll see something like:
HTTP/1.1 302 Found
Date: Sat, 08 Apr 2006 02:27:57 GMT
Server: Apache/1.3.34 (Unix) mod_auth_passthrough/1.8 mod_log_bytes/1.2 mod_bwli
mited/1.4 PHP/4.4.2 FrontPage/5.0.2.2635 mod_ssl/2.8.25 OpenSSL/0.9.7a
X-Powered-By: PHP/4.4.2
location: ./35595_keepintouch.mp3
Transfer-Encoding: chunked
Content-Type: text/html
That "Location" header is the URL it's redirecting me to.
Hope this helps!