I make my sophomores memorize some Shakespeare every year. I recommend that the boys memorize Sonnets 18 and 29.
I have never read 29 before.
Thank you so much for that insightful verse.
Its beauty nigh brought a tear to my eye.
But that isn’t “long”, is it? 
We also had to memorize poems, long ones like The Owl and Pussycat and Wreck of the Hesperus.
Ah!!! runs screaming.
I attended private (Catholic) school through grade 8, and public school after that. In public school I definitely had to memorize passages – mostly Shakespeare. I memorized lots of things in Catholic school, but I don’t recall literary works being among them.
One good thing you can do with memorized passages is create password lists, you know, when you have some lame employer who makes you change your passwords every three weeks – if you forget your password, you can always regenerate it knowing one of your past passwords:
Since1Golden
October2Declined
Into3Somber
November4And
…
(I’m up to “and mud”) 
It is if you’ve never had to memorize anything before.
Ideally, I’d like to hang out with the guy who memorized all of Shakespeare’s plays AND understood them.
Memorizing can be useful in some situations, though I say the ability to memorize something for varying periods of time is more useful than just having all of Shakespeare at your command. Being able to remember license plates, phone numbers, names, directions, or instructions can be very very useful to people.
Not to mention learning the actual techniques for memorization, rather than just what you memorized. These can be associations (remembering what part comes after what, as you mentioned, having to go through an entire passage to get to a relevant part, which at 26 years of age I still do with the alphabet sometimes), or using mnemonics (SPORTS - Slap Pull Observe Release Tap Shoot - the process for recovering from a misfire), or by putting them to music “Eight Six Seven Five, Three Oh Nine…” (Jenny’s phone number, according to Tommy Tutone).
You should have asked for people’s ages. Those of us who went to school back in the Dark Ages had to memorize all sorts of things, unlike kids today.
The only one that I still remember is Matthew Arnold’s poem “Dover Beach.”
Hey, hey, hey! Who are you accusing of having gone to school back in the ‘dark ages’, old timer?? 
I memorized the Hecuba speech from Hamlet.
“For what is Hecuba to he
Or he to Hecuba”
Public school, New England, mid-80’s. Absolutely. Not every teacher did this, but my third grade teacher made us memorize poems (I still can recite Frost’s “Stopping By Woods on a Snowy Evening”) and my ninth grade English teacher had us memorize bits from Shakespeare.
Impressive.
I had to memorize in English class. In my case it was Hamlet’s “How all occasions do inform against me” speech.
I have to say, memorizing poetry does give me a fair amount of pleasure (I’ve since memorized Browning’s “My Last Duchess,” Marvell’s “To His Coy Mistress,” the much-ballyhooed beginning of the Canterbury Tales, and so forth), and I have a pretty prodigious memory for things like stand-up comedy CDs and the like.
Agreed. In addition, memorizing things like Shakespeare sonnets and poems ingrains certain things like cadence, vocabulary, and a simple love of the beauty of the finely crafted verse.
To be fair, as somebody pointed out in a Shakespeare thread a while back, Shakespeare is best enjoyed watched, rather than read, as you’re basically studying his shooting scripts. It would be like, if in a few hundred years, kids were expected to read the scripts for Reservoir Dogs, Once Upon A Time In Mexico, and Star Wars, and memorize lengthy passages from those, instead of just watching a performance of said scripts like people were originally intended to.
I said “yes,” but it depends on your definition of long. 14 lines of Shakespeare seemed pretty long to me in seventh grade.
I went to high school from 1991 - 1995 and I am another that had to memorize the opening of the Canterbury Tales in Middle English. I want to say that it was my senior year, but I’m not 100% sure.
I never had to, and I must say that it strikes me as an awful idea. It’s the sort of thing I’d dream up if I were deliberately trying to design the worst curriculum possible.
I never had to memorize anything for my literature classes, but was required to memorize fairly large chunks of Bible (in Hebrew) in my Bible classes at my Jewish school. I hated it - I am a lousy memorizer, and I knew whatever it was would be gone from my brain as soon as I got out of the exam.