An important property of the derived category is that distinguished triangles don't just produce long exact sequences in cohomology. If A -> B -> C -> A[1] is an exact triangle and E is another object in the derived category, then you get a long exact sequence
... Hom(A[1],E) -> Hom(C,E) -> Hom(B,E) -> Hom(A,E) -> Hom(C[-1],E) -> ...
where these Hom-sets are sets of maps in the derived category.
A particular counterexample is as follows. We can view any abelian group as a chain complex concentrated in degree zero. There is a distinguished triangle as follows:
ℤ -> ℤ -> ℤ/2 -> ℤ[1]
However, we can take the last map ℤ/2 -> ℤ[1] in the sequence (which is not zero in the derived category) and replace it with the zero map. This still gives us a long exact sequence on (co)homology groups. However, if we let E = ℤ/2, then applying maps in the derived category from our new non-distinguished triangle gives us the sequence
... 0 -> Hom(ℤ/2,ℤ) -> Hom(ℤ,ℤ) -> Hom(ℤ,ℤ) -> Ext(ℤ/2,ℤ) -> 0 ...
where the last map is induced by the zero map from ℤ/2[-1] to ℤ, and so it must be zero. This sequence can't possibly be exact, and so the new triangle is not distinguished.