LCDs do have a refresh rate, but it is not anywhere near the important consideration that it is for CRTs. A refresh rate simply means how many times the image can be updated every second. So a 60 Hz refresh rate means that the monitor is capable of displaying 60 new images every second.
However, because of the way CRTs work, they actually flicker the images. That is, when the electron gun strikes a part of the screen, that part is bright, but quickly fades (< 1 ms). Then the rest of the 15 ms or so of that frame (frame = each time the screen is updated with a new image) is spent in total darkness. So you're basically looking at a strobe light. LCDs however have image retention, that is, the image sticks around until it's redrawn, so it's not flickering at all, plus it can spread the light energy over the whole frame rather than having to concentrate all that light in the first < 1 ms of a frame. In fact the current problem with LCDs is that the image sticks around too long and "bleeds" into the next frame, and hence causing all this ghosting stuff and people looking for low response times. Anyhow, so yes LCDs have a refresh rate; however, it's hardly a consideration for the LCD buyer. Instead, response time is the equivalent measurement, with the lower the better.
Oh and actually, for LCDs, gray-to-gray takes longer than black to white (except when the LCD is using overdrive, in which case gray-to-gray is faster). LCDs do better with big transitions because the voltage difference is larger; overdrive is simply doing a black to white (or white to black) transition then stopping it in the middle, when you've reached the desired (gray) state. If it's done properly.