Bud, on a car mixing tires isn't that bad. On a motorcycle it can be dangerous due to a number of items. The only part against that is keep a matched pair per axle, Dunlops or ?? up front and Michelins or Yokohamas on the back would be ok. Never mix and match tires on the same axle. Since, the same size tire from two different companies will not have the exact same dimensions. I'll use truck tire sizes for explanation purposes. A 31/10.50R15 tire supposedly is 31" tall, 10.5" wide 15" rim, but not all measure 31" some are 30.3 or 30.4 and some measure 31.5" when mounted. This can and will cause handling issues with the potential of being quite dangerous if mounted on the same axle.
The tire with the larger rolling diameter will have a greater load placed upon it at all times (probably more noticeable on a Viper or other stiff / full frame vehicle) the tires slip angles will be different making handling differences between left and right turns as the tire on the outside of the direction of curve will react differently.
The are so many things, that I can't explain beyond what I mentioned above. Try your local library for the book "How to make your car handle" by Fred Puhn published by HP Books
It has some good info on how tires affect handling. Its not the latest and greatest book, but very informative.
