"Yes, Earth will cross the galactic equator in 2012. As seen from the sun, the Earth does this every year– twice. However, the Earth won’t be physically passing through the plane of the Milky Way galaxy anytime in the near future.
Here's some background. The galactic equator is an imaginary great circle that divides the equally imaginary celestial sphere into two equal halves. The celestial sphere is– of course, a fiction. It’s the same fiction that so confounded the early stargazers, that, as seen from Earth, we appear to reside in the center of a great globe of stars. In modern times, the fiction of the geocentric view of the universe enables astronomers to use a workable coordinate system for mapping the sky."