One of the problems I think people have with pronouncing Feb-rew-air-ee is that they place a 3rd "r" in there, making it sound almost like the Scooby Doo "ruh roh" sound "Feb-ru-rare-ree" which of course is incorrect. It's actually not that hard to pronounce Feb-rew-air-ee if you make a mental note not to sound like Scooby Doo.
According to Charles Harrington Elster in his The Big Book of Beastly Mispronunciations. "The traditional and cultivated pronunciation, FEB-roo-ER-ee, is hard to say, and so most people say FEB-yoo-ER-ee because it is easier and because so many others do, not because it is right." and "As far as this book is concerned, however, the -yoo- variant may now be standard, but it is still beastly."
OUCH! I guess we better pronounce it right so as not be considered beastly! And the following should help us all in our quest for proper pronunciation of our favorite short month, FEBRUARY! (No, this is not me. I still screw it up.)