Randomly sending CVs almost NEVER works.
Phoning is better, because even if they don't have a job, they'll end up knowing who you are providing you don't call too often and you're friendly. So when hiring is approaching, you'll not only send the CV to the right person, but at the right time.
But if you really want to get chances:
1) Get a job dispatching/crewing/etc at an airline that DOES promote from the inside if the person's right for the job. EZY don't. BMI do. Eastern do. Do your reasearch! Most of the time it'll mean working for a third party, like Menzies or Serviceair. What matters is face time with captains. DO NOT try and shove your CV down their throat before you start knowing them. Be a bit patient. Once they like you because you're doing a good job, it'll be easy to get your CV put on the right desk.
2) Work at a small airport close to a big airport. Or even better, at the GA base of a large-ish airport (LPL, NCL, BRS, etc). Chances are, there will be some airline pilots who either own their own plane, instruct, examine, etc. Best contacts you can make.
Sending CVs and reading every aviation website out there will drive you insane very quickly.
P