The answer as you may have guessed is called a link cuff, and it is easy to accomplish. A link cuff is a single length shirt cuff which instead of having a button and buttonhole, has two buttonholes. Therefore you have the benefit of cufflinks without the bulk of double cuffs.
French cuffed shirts are the most formal of men's business shirts, and are only to be worn with a suit, link cuffs can be as formal (they are the only acceptable cuff for daytime formal wear), or casual as you like depending upon the shirt style you choose. The beauty of the link cuff variation is that you can break several mens style rules at once.
-You can wear cuff links with a blazer or sport coat. American stylists deeply frown on this. It is seemingly acceptable in Britian and Europe though.
-You can wear cuff links with a button down shirt. Absolutely no one sells button downs with french cuffs off the rack.
-You have the opportunity to create your own fashion style. You will never meet someone else wearing a link cuff. There is a price for being different though. It takes a bit of nonchalance, or sprezzatura, as the Italians say, to pull this off with aplomb.
So, how do you get it. Very simple! Take your favorite shirt to your cleaners and ask them to remove the buttons from your cuff and sew in buttonholes. Most will do it for a very small change. Soon, you'll be asking your tailor to make them especially for you. He'll balk at first, but eventually he'll come around.
These are dangerous times, its time to try something new.