I've cycled in Lisbon, and elsewhere in Portugal. Second craziest most dangerous place I have ever cycled, the absolute worst being just outside it - the road from Lisbon to Estoril. I have also cycled the length and breadth of Manhattan, and it does not compare to Lisbon, honestly. NYC is a walk in the park compared to Lisbon. Depending on where you are going the road surfaces can be quite bad, lots of stuff dug up or falling apart, and in places cobbles. Also the drivers are _not used_ to cyclists at all and will pass very fast and very close. So be careful.
On the upside the airport is very near the centre, so if you are bringing your own bike it is very easy to cycle into the city (it's pretty much a straight line.) If you are not cycling in, watch out for the taxis, you can get ripped off. Better get a taxi voucher from the desk in the airport, anywhere in the city for €12 or so.
I have not cycled in Rome but if I recall from the last time I was there there seemed to be a few cyclists getting on OK with the traffic.
Enjoy your visit to Europe!