Gums Bleed when I Floss in the Middle of New Permanent Crowns?

If you JUST got the new crowns, the tissue may be healing, so give it time.  If it has been a while, a week or two, then go back and see the dentist.  It is possible some contouring or clean up of cement is needed.

Bleeding gums and inflammation my be caused by residual cement at the gum line or just below it.I would suggest having the dentist take a look and possibly an xray if necessary.Removing the irritant usually resolves the issue. 
Happy Flossing!

It is hard to answer your question with knowing a bit more and seeing your teeth and new crowns.  How long ago were the crowns placed?  If just a few days ago, then might just be do to your gums settling down after the cementation procedure. 

If it has been a couple weeks, then I would suspect that there might be some excess cement that is still irritating your gums. Sometimes gums bleed because patients actually floss too hard, coming down on the papilla (gum between the teeth), traumatizing it themselves.

Improperly fitting crowns, fillings, veneers, etc., can also cause gum irritation.  Another reason is there might have been so much decay under the gums that the dentist had to place the crowns too close to the bone and gums.  This situation is relatively uncommon, but sometimes occurs, especially when patients wait too long to have their new crowns.

There should be a solution to your problem.  Go back to your dentist and see what he says before getting too worried about it.  He will no doubt make sure that the areas around your new crowns are clean and smooth and devoid of any excess cement and make sure that you are flossing/brushing correctly.