The diamond problem is the inability to decide which implementation of the method to choose. Scala solves this by defining which implementation to choose as part of the language specifications(read the part about Scala in this Wikipedia article).
Ofcourse, same order definition could also be used in class multiple inheritance, so why bother with traits?
The reason IMO is constructors. Constructors have several limitations that regular methods don't have - they can only be called once per object, they have to be called for each new object, and a child class' constructor must call it's parent's constructor as it's first instruction(most languages will do it implicitly for you if you don't need to pass parameters).
If B and C inherit A and D inherit B and C, and both B and C's constructors call A's constructor, than D's constructor will call A's constructor twice. Defining which implementations to choose like Scala did with methods won't work here because both B's and C's constructors must be called.
Traits avoid this problem since they don't have constructors.