I am not Russian but understand some. Out of all the Russian poets I've read I think Yesenin is my favorite.

I didn't see Russian on site (just English and Croatian). I found this with Russian versions, not with too many ads:

http://samlib.ru/w/wagapow_a/yesen.shtml

but the English translation is off. For example in "Letter To Mother":

http://samlib.ru/w/wagapow_a/yesen.shtml#a_letter_to_mother

"Ты жива еще, моя старушка?" is translated too literally as "Are you still alive, my dear granny?". Granny means more grandmother while in Russian it means more old woman, not just necessarily grandmother. So the translation from OP's link is better: "Are you still alive, my dear old one?"

You may be interested, if you're not familiar, with The Retuses, a Moscow-area band that has used Esenin's poems as the basis for several songs. Unfortunately, they were never recorded as part of an album, but many recordings are present on Youtube. Some are better than others, and the recording for Письмо к женщине is particularly well done: https://www.youtube.com/watch?v=UYgcUfMXJLw&t=2m30s

Шаганэ: https://www.youtube.com/watch?v=t8NYyVB_QyI Though I like this cover of it more: https://www.youtube.com/watch?v=TEN3Zugtqgs

Пой же, пой. На проклятой гитаре: https://www.youtube.com/watch?v=ME_Kxc9VlFw though I've sadly never found a better recording.

Mayakovski wrote a poem in honour of Esenin: http://m.poemhunter.com/poem/to-sergei-esenin/

I think HNers would also like Velimir Khlebnikov / Igor Severyanin / Vladimir Kamensky / Vladimir Mayakovsky and other Russian futurists of early 20th century.