You have four carburetor's that need to be in sync, one of many problems possible in just one carby will give you grief....
Choke maybe still open, allowing fuel to enter intake without proper air/mixture ratio. Or, the carburetor pilot jets and/or main jets are blocked or dirty.
Fuel contaminates do build up on the linings of various metals prior to reaching fuel reaching the carby's. Periodically (especially if your vary your fuel brand or even retailer's) clean the fuel tap filter and flush the carburetors with an aerosol carby cleaner.
Check condition of air filtration and while the covers are off, check all carburetor's to see slides and needles are moving in perfect unison when you twist the throttle.
The site link below shows an exploded view and name of parts for the carburetor's, Go up to header FZR600RE and click for entire bike in categories.
Cheers.
http://www.cmsnl.com/yamaha-fzr600re-1993_model9937/partslist/B-06.html