I still memorize things from time to time. My problem is that I have a very good short term memory but not such a good long term one. I can easily recite something with a closed book after a few minutes (depending how long it is) and will probably be able to recite it the day after (useful for exams) but a week later I'll be struggling and a month later I won't know it at all.
I do, just about, know God's Grandeur (though there are a couple of lines I'm never quite certain of) and one or two other things that I've known for years, but not many.
I did elocution lessons at school for a while, and we had to learn poems for that. I don't think we ever did in normal English lessons (UK, 1980's). Quotes from texts for exams, but that's not quite the same thing.