This is a super common problem with both generations of MR2, not sure about the MKIII.
If you know your suspension components or your steering is not the problem then it's either your wheels or tires. Some tires do not like to be balanced, I can't recall the brands/tire models but there are a few that are real crappers to balance, they're like lopsided ballons.
Also, you may have to use a Road Force balancing machine (Google it for locations near you) which is much more expensive but does a much finer job - you're almost assured of a good balance. I had this problem with all of my MR2s and the Road Force machine cured them all. And you have to tell the tech doing it that you need
zero balancing, none of this 'close enough' stuff. Not too long ago I had my turbo close to 140mph and it still did not have front-end shimmy (was kinda floaty though but that's another issue
)