I work in IT and it's always been "rowter". I believe you'd get laughed at if you called it a "rooter" in a computer setting.
As for route, well, personally it's "root" to me. I just couldn't sing "get your kicks on Rowt 66".
I've heard it pronounced both ways, depending on what part of the country I'm in at the time (same with "creek" creek/crick, "roof" roof/ruff, etc).
The only alternate pronunciation that drives me absolutely, completely, nails-down-a-chalkboard crazy is those pretentious idiots who mispronounce "Nevada". Especially the ones on shows like CSI who should theoretically know better.
It's Ne-vad(rhymes with dad)-a.
Not Ne-vad(rhymes with cod)-a.
Grrrr.
