First, thanks to everyone who has posted their experiences with tyres and handling here. The advice has been invaluable!
I bought my first Z4 last year and have loved every aspect of it except for the tramlining on the Lincolnshire country roads. Having taken it into the local BMW specialist, and to Kwik-Fit to get the suspension and tracking checked out, I found it very hard to believe that there was 'nothing wrong with it'! Maybe I was spoilt for the previous five years of convertible driving in my TVR? Maybe the 1-series which my girlfriend has is just not representative of BMW handling? (After all, the enjoyable drive in the 1 Series was part of the reason I went for the Z4). On nice flat tarmac it's great; around the twisty corners on the back road it is great; but trying to hold a straight line on a bumpy road was nothing short of dangerous. Having had the steering wheel repetitively almost wrenched out of my hands, I was ready to give up.
Anyway, I was on the verge of selling it and buying something decent to drive when I started doing some research. Clearly I was skeptical about how much difference some new tyres will make to the ride and handling of a fundamentally flawed car. But I thought I would give it a go rather than sell the car. Having called the insurance company to check they were happy with putting any tyres on (they were - Esure), I set about finding a dealer who could supply some decent non-runflats to replace my original Bridgestone Potenza RFTs. This wasn't as easy as I thought - some seem to think that your car will explode and everyone will sue everyone if you get a puncture and/or have a smash. Too many 'accident "helpline"' style lawyers have obviously got us on the back foot - what a shame.
However, Selecta Tyre in Newark-on-Trent were great. They were helpful on the phone, and for around 80% of the Kwik-Fit price, fitted 4 Michelin Pilot Sport PS2s to my Z4. I drove away from the garage on Wednesday afternoon in what might as well have been a new car. The ride was better, smoother and more refined. The handling in every respect was improved - more responsive and accurate, and most importantly of all, no more tramlining or snatching of the steering along less-than-perfect roads.
Whilst I cannot compare the Michelins with any other tyres than the old Bridgestone RFTs, I can whole without hesitation recommend to anyone who has a Z4 to fit non-runflats. £760 is not a small amount for new tyres, but it was worth every penny! My only regret? I didn't do it sooner!
If you've got a Z4 and don't like the handling - get rid of the runflat tyres and stick some foam in the boot. It might one day be slightly more of a pain if you do get a nail in your tyre, but compared to the better driving experience, well worth it!