Check out by setting a Cookie through JavaScript that will hold the client's timezone. It's rather lenghty, but that is because it is quite verbose. I've implemented a solution along these lines in one of my own apps and it works quite well.
You could also send the timezone via Ajax to the server and have it do whatever you need to do it with then. Or, if you are not doing any serverside calculations with it, just apply the timezone client side where needed. Really depends on your usecase.
In addition to that, I suggest you let the visitor set his timezone himself and store that in the Cookie or a Session.