I would definately take your own bike and not rent one. We went to France for 2 weeks biking last year with our own bikes. Being on a nice bike that would meet all performance expectations was great. There was never any frustration about having a clunker, poor fit, or poor gearing, etc. Good cyclists who rent are usually disappointed with their bike.
We flew British Air to Geneva, Switz., through London and there was no charge for the bikes. We were able to leave the bike boxes for nearly 2 weeks (no charge) at our arrival/departure night hotel.
My second recomendation is travel light! We had a daypack and fannypack as carry-on and the only checked luggage was the bike. This worked well for "credit card" travel in early June, even in the Alps where we saw snow in the air at 9,000 ft.
We do Italy in May.