A balance has to be struck in this situation. If a pilot asks for a turn to avoid weather and ATC refuses, he/she must make a decision whether to continue as cleared or to take responsibilty for a turn. Why waste precious time by asking again when you have already bee told "No". As anotherthing says, if you have been refused a turn it is for a bloody good reason, i.e. there is another aeroplane close by and if you take the turn you want, you're going to lose minimum separation.
As a pilot and a controller, in this particular scenario working as a controller I have used the following procedure. It may not be listed in the book but everyone was crystal clear.
Working TMA North last September when thunderstorms appeared to be a daily occurence, I had an aircraft outbound from EGSS on a CPT deaprture. The pilot requested a 30 degree right turn which I refused as there was traffic 3 miles northwest at the same level. The turn would have resulted in a loss of seapration hence my refusal. The pilot prevailed as was his/her right to do so, so I made it perfectly clear with something along the lines of "Callsign, traffic 2 o clock 3 miles same level, right turn will put you in conflict with it, if you choose to turn right maintain your own visual separation".
I continued with regular traffic information until the conflict was clear. They were flahsing white on my screen and I was quite frankly shi**ing myself but it worked out in the end and I wasn't held responsible. Quite luckily, even though they were reasonably close, minimum separation was maintained throughout. Through pure fluke mind you.
The downsides to the situation were the massively increased workload due to having to monitor it in a very busy and complex environment but we all kept our noses clean which must be a bonus?
I can understand the situation from both sides of the fence and I think the only way to resolve our differences (read: misunderstanding of each others jobs) in this situation is familiarisation of each other. Fam flights and visits to ATC are absolutely essential if we want to bridge the gap in knowledge.