Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs
I think there is a number of hard problems that become easy for triangle-free graphs; especially those deal directly with triangles such as Partition Into Triangles (Does G have a partition into triangles?). Other less trivial examples include:


Stable Cutset Problem (Does G have an independent set S such that G-S is disconnected?). See:  On stable sutsets in graphs, Discrete Applied Math. 105 (2000) 39-50.  
Intersection Graph Basis (Is G the intersection graph of subsets of a k-element ground set?). See: Problem [GT59] in: Garey &amp; Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.