It has been conjectured that L-spaces are equivalent to 3-manifolds with non-left-orderable fundamental group. Supposing that this conjecture is true, some interesting (perhaps even surprising) behaviour is suggested both for L-spaces and for left-orderable groups. This talk will outline some of the supporting evidence for the conjecture, and then discuss some calculations in bordered Heegaard Floer homology for studying a particular family of graph manifolds that do not admit taut foliations. In particular, we'll give examples of what might be termed 'Heegaard Floer homology solid tori'.