I had Bridgestone Potenza's on my Dooper Convertible and to be honest the ride did not bother me, yes it was hard but I quite like that.
When I needed two new ones I went for Hankooks. I had only replaced two and the new ones went on the back and the car handled like a bag of sh**e. When I went back to the tyre place I mentioned it to then and they said it was because of the flexibility in the tyre wall. The got me a Hankook and a Potenza off the rack and got me to press down on the sidewall. The Hankook gave a fair bit but the Potenza's were almost solid.
I needed a new front tyre as I had been given an MOT warning on one of them so I replaced the front two with Hankooks and the handling improved considerably.
I am happy with the Hankooks, but if I had known what I do now I would have stuck with the Bridgestones, avoided the bad handling and continued to enjoy the bone shaking ride.