Omar Khayyám is by far the best-known Persian poet in the English-speaking world, due to Edward Fitzgerald's famous translation of his Rubáiyát. Although he's also well-liked in Iran, the likes of Ferdowsi, Sa'di, Hafiz and Rumi are usually considered superior, although almost unknown in the English-speaking world. Most of them were very well-known in France and Germany during the 18th and 19th century, and inspired several German poets. But since English translations couldn't really carry on, these poets were forgotten after a while.
In an interesting historical usage of this trope, while Edgar Allan Poe is considered to be a great writer today in the US, or at least he's acknowledged for his poetic merit and his creation of the mystery genre. During his lifetime he was much more popular in Europe than he ever was in America. The very well-received French translation by poet Charles Baudelaire helped a lot and is still the go-to French Poe translation over a century later. Some literary scholars argue that Baudelaire's version of The Raven is superior to the original. To receive this level of praise is extremely rare for any translation.
Lord Byron. He was rather popular in Britain in his lifetime, but his fame always had a scandalous tinge to it, and his work was often dismissed as being more in-your-face provocative than good. In modern critical appraisals, Byron seems to be overshadowed by Shelley and Keats. On the continent (especially in France and Russia), Byron was the face of English Romantic poetry and the possibly best English writer ever, second to only William Shakespeare. To say nothing of Greece, where Byron is still literally venerated as a national hero (it helped that he literally died fighting for them even though he had absolutely no compelling reason to do so).
William Shakespeare's sonnets are popular everywhere. But which one is the most popular or most well-known? "Shall I compare thee to a summer's day?" Not a chance! Czechs will invariably name Sonnet 66 as their absolute favourite, although it might mean it's the only one they know, but still. This sonnet has been translated gazillion times, and it includes both brilliant published versions and fan-fic-like translations of various quality. One musician wrote a melody and has a song version of it. Czech Professors of English Literature give lectures just about this single sonnet, or if it's about Shakespeare's sonnets in general, there is always a special discussion of Sonnet 66. It keeps receiving Shout Outs in novels and theatrical performances. Czechs sure do love their Crapsack World. Possibly due to similar reasons, it was also quite popular with the Soviet intelligentsia.
Similar to the above William Butler Yeats is well known across the English speaking world, but in America at least The Second Coming is incredibly famous and endlessly referenced in popular culture. Irish people are often bemused by this as in his own country The Second Coming is not considered any more notable than any of Yeats' other poems - less so in fact as poems like Easter 1916 have a political context that make them much more likely to be taught in schools and others like Sailing to Byzantium are considered more beautiful.
Robert Burns has been very popular in Russia since the Imperial era when we was considered the "People Poet".