Well, any finitely generated group G of exponent 3 is finite by a classical theorem of Burnside. And since the order of every element is 3, the order of G must be a power of 3 by Cauchy's theorem. It follows that G is a finite nilpotent group. A similar argument shows that the same is true for any finitely generated group of exponent 4. This is unknown for 5, and false for 6.
Correction: it seems (see the answer of Primoz above) that any group of exponent 3 is nilpotent, altough it can be infinite if it not finitely generated.