Hi Tracy -
Did you have this in writing ie as part of the rental contract, when they moved in 3 years ago, not that it changes anything now!
You should always ask your tenants months before a contract is up, if they intend to stay or move out and as of when!
The laws seems to be on the tenants side in France not the owner, I have heard many horror stories, and for that very reason we would never rent out in France, but that's us. That aside, as it's not about us, it's about you! You are lucky to have had 3 years constant tenancy, and hopefully without too many problems, and the fact that they want to move out, as in lots of cases, people do not want to move out, and getting them to move out seems very difficult - and without them paying any rent!
Yes, I would advertise as of yesterday, and make sure that under the terms and conditions, you point this out to your future tenants ie and get them to sign/initial that particular part of the contract, and if they are not willing to do this, well, I would not have them move in!
Good luck, hope all works out for the best...
PS! Why don't you tell us on SFN, AngleInfo etc where you are in France, where you are near + what size property you have available to rent, + furnishing, minimum rental period, if anything is included, is car needed, it's a long list, but only to cover yourself etc - word of mouth sometimes works better than any ads, and it might just be that someone happens to know someone who needs a roof over their heads... :)