1. E. Kowalski
   PhD thesis, Rutgers 1998
   http://pws.prserv.net/frog-parrot/articles.html

2. K. Soundararajan
   "Non-vanishing of quadratic Dirichlet L-functions at 1/2"
   Annals of Math. 152

3. Perelli & Pomykala
   "Averages of twisted elliptic L-functions"
   Acta Arith. 80 (1997), 2
   This one does quadratic twists of a fixed modular forms.
   Doesn't get positive proportion; it's still unknown in that
   situation.  Simpler than Soundararajan, which is the strongest
   known, but Soundararajan is not for quadratic twists. 
  
4. Iwaniec, H.
   "On the order of vanishing of modular L-functions at the critical point"
   J. de Th. des Nombres de Bordeaux 1990
   (This might be tricky to get ahold of, but Emmanuel can easily get it.)