Unfortunately, most landlords and rental agents are fairly unimaginative and pretty much insist on the 3 months of back payslips (i.e. the last 3 months, not just any 3 months) or paying 6 to 12 months up front.
One approach some folks use is to take a short-term "vacation" rental for 2 or 3 months on arrival, and then use that time to look for long-term accommodation. By the end of the 3 months of "temporary" housing, you should have the necessary 3 pay slips and you can find a long-term place with somewhat less hassle.
You're going to need to establish a French bank account when you get here anyhow. (Many landlords insist on setting up the rent as an automatic debit from your local account.) Just don't close out your US bank account, and check before you leave on how to best set up and authorize a wire transfer to your French account.
They normally won't let you do international wire transfers online, but with some pre-arrangements you may be able to set things up and then "authorize" the actual transfers as you need them. And besides, you'll need your US account when you go back to visit. Also, hang onto your US credit cards - they come in handy. You can just change the billing address to your address in France.