This is indeed the Red Army Marching Song I learnt in high school (though I only memorized the first and third of Stewart's superabundant stanzas). The Fireside Book of Folk Songs had a singable English version in it, I think. The Cyrillic lyric Sorcha linked to is related, but not the same (too many extra "Ekh's" at the beginnings of lines (there should be only one in the first stanza - not on the opening line, either - and two per stanza thereafter, as in Stewart's transliteration.) Maybe I'll put it on my website in Cyrillic, with the Esperanto translation.