I started off by posting a series of comments scattered all over the page, but I thought I should sum them up in a standalone answer.
Generally speaking, there have been similar shifts in many other languages. And they are even happening right now as we speak.
But first things first. Since you mentioned Slavic languages, as the most obvious example, in Old Church Slavonic, an o was an o. In contemporary Russian, it can be anything from a schwa to an ʌ to an ɔ, depending on the position relative to the stressed syllable (e.g. молоко, milk, /məɫɐˈko/ or /məlʌˈkɔ/; водоворот, swirl, /vədəvʌˈro̞t/). Also, in Old Church Slavonic, there were a number of nasal sounds, which are absent in pretty much all contemporary Slavic languages with the notable exception of Polish.
Secondly, don't get me started on German. If you don't know how to correctly pronounce Soest, Troisdorf, Huonker, Pankow, Laermann, Hueck, you will pronounce them wrong. It happens to native German speakers all the time.
Speaking of Germanic languages, the most notable vowel shifts happened in German and Dutch (Wikipedia even mentions them in the article on the Great Vowel Shift). It's just that there was at least some concerted effort to keep the spelling consistent with the (changing) pronunciation. The pronunciation shifts were accompanied by spelling shifts, if you will. Hence the popular but wrong assumption that there weren't pronunciation shifts to begin with.
In other words, what made the pronunciation stray so far from spelling in English was not the Great Vowel Shift; it was the absence of the accompanying Great Spelling Update.
Now, it's always a bit harder to explain the absence of something rather than its presence, though one of the other answers does provide an interesting link. On a more general note, I will say that spelling reforms are the domain of politicians, one of the most prominent and recent examples being the German orthography reform of 1996, kicked off by the Conference of Ministers of Culture and later monitored by the International Commission for German Orthography. English, however, traditionally lacks such regulatory bodies.
Anyhow, vowel shifts happen all the time, especially on the dialect level. Now that I have zeroed in on German, I'll just take Bavarian as an example. In Bavarian, viel is not pronounced as /fiːl/ and ein Haufen is not pronounced /aɪ̯n ˈhaʊ̯fm̩/. But again, there is some effort to keep the spelling consistent with the pronunciation, so if you came up with the crazy idea to write a Wikipedia in Bavarian, you would spell viel as vui, vei, vii or fui, and ein Haufen as a Haufa, to reflect the actual pronunciation. And there are also Wikipedias in Ripuarian, Plattdeutsch, Alemannic... It's hard to imagine, say, standalone Australian, Canadian or Texan Wikipedias where the spelling mirrors the local dialect in such a manner.
I guess I can sum these ramblings up as follows: vowel shifts happen all the time. Spelling conventions are a question of politics and culture.
