Sling Media announced that a SlingPlayer Mobile client for BlackBerry devices would be coming out as a public beta and it is now available for download. Visit mobile.slingmedia.com/ from your BlackBerry browser to install it since there is no desktop installer for the software. Officially supported devices are the Pearl 8120, Pearl Flip 8220, Curve 8320, BlackBerry 8820, Curve 8900, and Bold 9000. I understand that the EDGE-only devices will also work, but performance won't be as good as you will find on a 3G BlackBerry device. You also need to have BlackBerry OS 4.5 or later to run the software.
Sling Media usually has these public betas up for a month or two while people test it out and then make the official client available for US$29.99. I don't have a SIM provisioned for the BlackBerry right now so I cannot test this client out, but I am sure reviews will start appearing very soon.