Are there Graduate schools that do not require GMAT if you have a good undergraduate GPA average?

Some internationally accredited programs which do not require GMAT are Durham Business School (UK), U. Carlos III Madrid (Spain), U. Edinburgh (UK), U. Exeter (UK), U. Glasgow (UK), Henley (UK), Hull U. (UK), U. Kent (UK), Kingston (UK), Solvay (Belgium), TiasNimbas (Netherlands-Germany), Toulouse Business School (France).

the Chicago U

1) University of the WEST

2) Johnson and Wales University

Actually, there are. It's not uncommon to see the undergraduate school you went to give you a pass if you have a high enough GPA.

Also, some graduate schools accept you right away in some specialities; such as any area in science.

The school that I attended for my undergrad did not require me to take the GMAT in order to attend its grad school. Other than that situation, grad schools require the GRE or GMAT, etc.

Edit: The only exception is the EMBA (Executive MBA). Many business schools offer a waiver.

Since so many schools have inflated grades or have lowered their standards, standardized testing is a very important factor in grad & professional schools admissions. You are unlikely to get into any grad/prof'l school without good scores on the GMAT/GRE/MCAT/LSAT, as required.

All colleges now require some sort of testing such as SAT's.

If you have a learning disability, you may not have to participate in the SAT. Talk to your high school guidance counselor.