The first poem of the episode was composed jointly by Kinto and Sei Shonagon. According to the Pillow Book, this was a fairly common game, especially between the men and Shonagon, because she so frequently wrote eloquent and unexpected replies. This poem and the story about it is actually recorded in Shonagon's Pillow Book:
Sora samumi
Hana ni magahete
Chiru yuki ni
Sukoshi haru aru
Koko chi koso sure.
"Snow falls like flower petals from the wintry sky. For but a moment, I feel the coming of spring."
The poem above is based on the Japanese version of another Ba Ju Yi poem, which is called, "Southern Snow," and is roughly translated in the show as, "Three in the morning, snow in the cold air, February in the chill of the mountains, I feel the coming of spring." The lines which Kinto sent Shonagon refer to the last part of this poem. Shonagon recognizes his intent, and begins the poem with 3 lines that refer to the earlier part of the poem, but she also artfully compares snow falling to cherry tree blossoms falling. Starting to understand why she's such a legend yet?
Kinto's next poem about the waterfall is #55 in the Hyakunin Isshu. To me, there's a lot of hidden meaning ensconced in what seems like a simple nature poem, things like the legacy of a person after death (the theme of the whole episode):
Taki no oto wa
Taete hisashiku
Nari nuredo
Na koso nagarete
Naho kikoekere.
"No more can we hear the sounds of the waterfall, but its legacy continues to spread."
The very last poem is attributed to Empress Teishi, and as discussed in the episode, it was not made a part of the Hyakunin Isshu. The episode hints that Teika did this to preserve the image of Empress Teishi given in Shonagon's Pillow Book, but it's one of those things that is hard to know. Also there's a lot of discussion about what Teika did/didn't put in the Hyakunin Isshu, which I think is ultimately not useful. We have to remember that he wrote it for a relative, not really for the type of publication it has come to be. Anyway, this poem is beautiful and dignified:
Yo mo sugara
Chigirishi koto wo
Wasurezu wa
Koin namida wo
Iro zo yukashiki.
"Nights of love unforgotten in mourning tears of a curious hue."
Which I think is more gracefully translated to something like, "After all this time we spent together, I would like to see the color of your tears after I die." T.T