In Portuguese we have "ontem" for yesterday and "anteontem" for the day before yesterday, much like in Ukrainian and Polish. We also say jokingly "antes de antes de ontem" for three days past today, or literally the day before the day before yesterday, but I don't really know for sure if this is good portuguese, it's more like a regional idiom :P
Yes, "a/the day before yesterday" is really long and for Ukrainian and Polish (and I think for other Slavic) speakers it's strange to don't have word for позавчора/przedwczoraj (1st is Ukrainian and 2nd is Polish).
Korean has them all. 그끄제, 그제, 어제, 오늘, 내일, 모레, 글피, 그글피 means -3, -2, yesterday, today, tomorrow, +2, +3, +4 respectively.
German as well: gestern is yesterday, and for each day before that you put a "vor" at the beginning. Vorgestern, vorvorgestern...but more than one is really uncommon.