Add Olivier's Henry V as well, another terrific Criterion DVD with a highly informative commentary track.
Mazursky's Tempest was in HDNet Movies rotation a few months ago, though I don't think they're currently showing it. The bits I caught of it looked intriguing.
I took a class on Shakespeare in university, and in one of those classes, there was a discussion about film adaptations of Shakespeare's work. When I mentioned Branagh's "Much Ado About Nothing", the teacher immediately claimed that he thinks it is the best film adaptation of a Shakespeare play ever. I completely agree. The only bad thing about it is Keanu Reeves, who gives a very weak performance as the villain. I watched The Taming Of The Shrew (1967) starring Elizabeth Taylor and Richard Burton for the same class. They were a little too old for their characters and Taylor could not handle Shakespeare's language nearly as well as Burton did, but it was still a good movie overall. I also saw a clip from Branagh's "Hamlet" and was amused by the fact that it was of a sex scene, which was not in the book, as I assumed that Branagh only put it in the movie so that he could have one with Kate Winslet.
I highly recommend Grigory Kozintsev's Russian adaptations of "Hamlet" and "King Lear", though the Ruscico DVDs that are available in the U.S. are a bit pricey and not high-quality.
I believe The Virgin Spring is actually based upon a 13th Century Swedish ballad, and does not carry any Shakespearean roots. Still a darn good movie though.
Anyway, I don't believe this discussion could be complete without at least noting the occasionally wonderful BBC presentations of all 37 of Shakespeare's plays. The series, while uneven, produced some fantastic adaptations and featured performances from the likes of John Gielgud, Anthony Hopkins, John Cleese, and Patrick Stewart. Certainly not high budget productions, but usually solid nonetheless.
Another of the "looser" adaptations worth noting would be Gus Van Sant's My Own Private Idaho. Starring River Pheonix and Keanu Reeves (yes, I know, bear with me) as two street hustlers, the film borrowed plot elements and dialog from Henry IV Parts I & II. It's an interesting film, to say the least, though not for everyone's tastes.
Branagh's Love's Labours Lost is also a somewhat interesting film. Though definitely the weakest of his adaptations, it's a pretty entertaining musical reinvention of the play, featuring old musical standards inserted into the original text. Who can complain about watching Nathan Lane singing "There's No Business Like Show Business?" Perhaps it may feel a little out of context, but...
John Cassavetes made a rather eccentric adaption with his TEMPEST [1982]. Raul Julia does a fairly outrageous turn as a Caliban-figure. ("I want to balanga you with my bonnie-johnnie!").
Gielgud in 1964 made a straight-forward TV version of HAMLET as then being played at the Lunt-Fontanne Theater on Broadway, with Richard Burton in the lead. There's simply no better Hamlet than Burton's. The DVD is available, luckily.