We always cut our poppies down to the ground after flowering, the perennial Oriental hybrids that is. The other annual types we leave a couple of head to self seed and pull up the rest as they will not flower again.

I do the same as Owdboggy.

Cut them off if you don't want a garden with just poppies. You can leave just a few to self-seed, as per Owdboggy, but don't be too generous!

There was a time when the dried seed heads were a popular decoration. You need to keep an eye on them and cut them just as they are beginning to ripen from green to brown - bring indoors to a cool-warm dry place and hang upside down.
Keep checking them and as they dry, loosely fasten a paper bag around the pods to catch seed which can be used in baking or scattered in the garden if you decide that you would like more poppies after all.