I have been to the Dolomites three times and never really been disappointed.
Before late May some of the higher road passes can still be shut. The skiing season is all but over and you can sometimes end up in the closed season, between the end of the skiing rush and the start of the summer tourists. The same can happen if you go much after the middle of September.
Many of the big lifts stay open during the summer months, carrying tourists up to the mountain tops, so (if you don’t want to ride you bike everyday) there are other things to do.
I would think about July into August. OK, it can be a little bit more crowded but you will not find yourself rattling around in ghost towns. Late August, say after the 20th, for a two week period would probably be my best choice. The Europeans are back at work and Les Brits are probably clearing off back to the UK.
There are some very good websites around (including UKGSers) with lots of tips on what to do and where to go. It’s a great part of the world, quite different to the Alps and the rest of Italy. There is a really good book (sorry I am at work, so can’t remember the title) written by an American, detailing all the best routes and passes. See if you can get a copy – I will sling the title up when I get home.
Or PM me if you want a photocopy of the best pages. I have also got a good touring map I could copy for you. That alone would make it paying the £12 annual fee to be a subscribed Tosser.
PS Not sure that any of the French holiday bits are shut in August? Paris offices, yes. Everywhere else, buzzing.