I went in the dry season, at the very end of June.
Yes, you don't get that mirror effect, but it is still absolutely stunning.
The 3 days tours take in much more than just the salt flats themselves - you also get lagoons, hot springs and the Atacama.
Besides, during the rainy season parts of the tour can get cancelled (for example Fish Island). Which would be a shame to miss.
Having travelled extensively in South America I would say this was without a doubt the highlight.
One piece of advice - do a fair whack of research into the companies. There are plenty of threads about that. The rumours about heavy drinking drivers on some of the companies is true.
Just don't cheap out on the company. It's not worth it.
And #6 is right - It will be incredibly cold at night at that time. We reached about -6C, but it felt a lot, lot colder. It gets even colder in August. But, as long as you bring enough clothes, you'll be enjoying yourself too much too care.