I've heard it's a bit hit and miss and if you don't have a good lawyer working on your side getting through might be tough. I'm not sure if it's a different story for H1B Visas though.
In a nutshell, I've been told time and again that having a degree will help, although how much one that isn't related to your discipline may not be worth much more then the paper it's printed on. To give my experience, I've been told by multiple companies within the last year that it's basically no Comp Sci (or equivalent two year college course), no chance. Your degree in an unrelated field might help a bit, but I wouldn't put much weight in it when it comes to immigration VISA time.
Everything I hear sounds so anecdotal it's hard to get a grip on what matters. Hopefully the more applied years helps too (I have 6). Still, it sounds like getting in may be a matter of strong-arming through law/lawyers.