As the title states, how do I pronounce the word "Linearly"? I did some Google searching on this but I was not able to find any guidance.
The CMU Pronouncing Dictionary gives an American English pronunciation for linearly: http://www.speech.cs.cmu.edu/cgi-bin/cmudict/?in=linearly&stress=-s
L IH1 N IY0 ER0 L IY0
In International Phonetic Alphabet (IPA) this would be
In the American Heritage Dictionary-style pronunciation respelling it would be
In the Wikipedia pronunciation respelling style it would be LIN-ee-er-lee
As has been discussed in other answers, in standard non-rhotic British English, the 'er' of the third syllable is pronounced non-rhotically—that is, indistinguishable from an ordinary schwa. However, there are a number of rhotic British English dialects with a substantial number of speakers which would pronounce the 'er' rhotically.
/'lɪnɪəlɪ/ (depending on your dialect of English). Four phonemes (sounds), anyway: LI-nee-uh-li.
In fact, there is some pronunciation help online: http://oxforddictionaries.com/definition/english/linear?q=linearly#linear__9 and it looks like ODO agrees with me as far as British English is concerned.
HERE is one of those free MIT on-line lecture courses. This one is on linear algebra. It should use the word "linearly" quite a lot, I expect. In particular, try Lecture 9.