This is significant for the Rangers, of course. They've been pursuing him for several years, and while Greinke has a no-trade clause, the Rangers are not one of the teams he can block a trade to.
The Royals' asking price for Greinke has been exorbinant, but with Greinke now forcing the issue, they may feel that they are backed into a corner. With Greinke under contract for two more years, they could simply say, too bad, we aren't dealing with...but Greinke has walked away from baseball once before, and I wouldn't find it shocking if Greinke decide that he simply wasn't going to play for Kansas City anymore, and refused to report if he wasn't dealt.
Alternatively, they can go to spring training with a young, rebuilding club whose superstar pitcher is going to be unhappy and not making any effort to hide it. Kansas City may be best served by simply finding the best offer they can right now and moving forward.
Greinke was a disappointment last year, on the heels of his Cy Young Award winning 2009, and there was some suggestion that the hopelessness of the Royals' situation, and the futility of pitching for them, had finally worn on him to the point that he wasn't performing to capabilities.
The Rangers, with Cliff Lee having signed with Philly (*cries*), are doubtless going to continue pursuing Greinke, and have pieces to get the deal done, even if they refuse to part with Martin Perez. Jurickson Profar is supposedly a sticking point, but I can't imagine the Rangers would make him untouchable as part of a deal for a pitcher like Greinke.
So, we shall see...this is definitely a situation worth monitoring going forward...