It's my daughter's name, and we pronounce it lin-AY-ah.
Linnaea borealis is a native flower here in the NW US, and all my botany professors in college pronounced it lin-AY-ah, so that is how I've always pronounced it.
A lot of people here say lin-AY for some reason; (there's an "A" at the end of the name!) obviously they are not familiar with the name.
My Spanish friends pronounce it lee-NAY-ah, and I've heard it pronounced lin-AY-ah and lin-EE-ah by Brits. I've never heard the Swedish pronunciation.