Every New Year's, when your parents used to put on Ирония Судьбы and make you watch it, you heard his poem at the beginning:
Со мною вот что происходит:
ко мне мой старый друг не ходит,
а ходят в мелкой суете
разнообразные не те.
They also somewhat butchered a lyric poem of his by turning it into a running joke in Забытая мелодия для флейты:
Весенней ночью думай обо мне,
И летней ночью думай обо мне,
Осенней ночью думай обо мне,
И зимней ночью думай обо мне, обо мне.
There are dozens of other poems of his that were turned into songs (just consult his Russian wikipedia page for a full listing; the most popular was certainly Хотят ли русские войны).
I would highly recommend reading all the authors of the Шестидесятники generation - Akhmadulina, Rozhdestvensky, Voznesensky, Yevtushenko and Aksyonov. Brodsky is a bit of an acquired taste but also worth reading, and of course Okudzhava and Vysotsky should be listened to.
Of course, I also got into some of the writers that don't fall into that generation but are more recent - Boris Vasilyev, Valentin Rasputin, Yuri Trifonov and Vasily Shukshin.
Not only that, but Russia still seems to be producing good writers even now, like Boris Yekimov, Vladimir Makanin, Yevgeni Grishkovets and Vasily Belov. Pelevin and Akunin are both a bit low brow but obviously popular.
Agreed on all counts. I will look at the authors you recommended here and pick up one of them shortly. Right now I am mired in Volkov's St. Petersburg: A Cultural History which is doing an interesting job of detailing the artistic culture of the city across the centuries. Though it's quite dense.