It’s regional in the U.S. Around here (Silicon Valley) it rhymes with “out.” Where I come from (New England), it rhymes with “boot.” I still say it as I learned it; but I find that I do have to say Cisco makes rowters (rhyming with “shouters”), not rooooters (rhyming with “hooters”).
The question becomes even more interesting when you add in “root.” I grew up saying “route” and “root” exactly alike. But in the Midwest, where I went to school, “root” rhymes with “put.” I don’t think anyone pronounces “route” like that “root.”