"R" was the sound that gave me the greatest trouble when I was first learning French. I received varying advice when I asked for pronunciation help: my American-born French teacher in high school suggested that I practice gargling; authoritative classmates who were native speakers of Russian and Spanish gave conflicting recommendations on where to position my tongue.
Through trial and error, I've found that I can approximate a French "R" as one might hear on the streets of Paris or on RFI by attempting to enunciate an American English "L", then rapidly switching halfway to an American English "R" sound. The trick lies in bringing the tip of the tongue close to the region behind the upper front teeth without actually making contact with the palate. This gap is needed in order to produce vibration. As air passes over the tongue, the sound that emerges from the back of a throat will probably sound like a cross between an English "R" and the Russian "KH" Mercuryblues describes above.
Depending on location, the French "R" might be trilled like the Russian or Spanish "R". As a spoken language metropolitan French hasn't been around for all that long; a century ago less than a third of all Frenchmen living within the "Hexagon" could speak let alone write standard Parisian French.