Personally, I try not to speak of it but if I’m ever in the unenviable position of having to utter it’s unholy name I usually say may o NAZE a la Louis Gossett Jr. in An Officer and a Gentlemen.
Pretty much MAN-aze
What other pronunciation is there, other than with more of a french accent?
What SeaDragonTattoo said. Two syllables instead of three.
My grandmother used to call it MAY-naze. I use the first option in the poll. Or I just call it mayo.
I say MAY-uh-naze (/ˈmeɪ ə neɪz/). Definitely three syllables for me, with the second being a pretty short schwa.
Just like Louis Gossett, Jr.
MIracle Whip.
Same. Midwesterner.
Oh, and for the record, when I’m not over- pronouncing it like Mr. Gossett Jr., I use the “proper” pronunciation with three syllables.
I’m in between MAY-uh-naze and MAN-ase but I hate it so I chose the third option. I pronounce it sort of like Neil Armstrong pronounced “That’s one small step forrrrr man”. Sort of like “Maaaan-aze”
may o naze
UK, for what that’s worth.
I guess I’ve heard MAY-naze, but not often. May-0-nayz or mayo - CA and Nwingland.
May O.
The Nays have it.
Daylight come, and me wanna go home! 
I give it the Irish pronunciation: May O’Nais.
Like this, only with the accent on the last syllable – may-uh-NAZE. Everyone I know to speak to pronounces it like this, I think (we’re British [English variety]).
may uh NAYS
I’ll say MANaze (“Pub burger, please, no MANaze”), or mayo (“Sandwitch of the day, please. You don’t put mayo on that, do you?”) but I’ll never eat it because it is terrible, terrible nasty stuff.
Long ago on the SDMB, I learned that mayonnaise is the devil’s semen. Therefore I picked the third option.