In the one draft where I went in with a specific strategy, that strategy worked out extremely well. The goal was to take best available players for the first couple rounds, interspersed with a few picks that attempted to start runs at positions, with the goal of starting a run, then changing that run when the pick came back to me, with my pick being the start of each new run. It worked surprisingly well:
First two rounds went best available, with Helton and Sheff coming my way. By the time things got back to me (9th slot) for the third round, only Santana and RJ had been picked. Noticing everyone going hitter, I said, ok, let's give it a shot- took Sheets as the 3rd overall pitcher. Immediately, the next picks were Prior, Oswalt, and Pedro. No big surprise- all solid 3rd round picks- but it said to me that I might have just started a nice run. Of course, those guys didn't take pitchers on the way back, so I went with the next obvious position to start a run- SS. Took Mike Young as the 3rd SS taken (Jeter was taken early). This is when the strategy paid off- in the next 16 picks before my 5th rounder, 10 of them were either starters or SS. I feel like by starting the pitching and SS runs back to back, I was able to get effectively the "first pick" in an "all pitcher" and an "all SS" round, plus the first pick in the "next round of the rest of the draft."
Continuing like this when I noticed a dry spell in picks at a certain position, I was able to be at the start of runs at SS, two separate runs at SP, and upper tier (but not elite) closer, as well as a mini-run at kinda-good outfielder. Coupling that with grabbing a few guys I had an eye on (AJ Burnett in the 9th, Ramon Hernendez in the 15th, Chone Figgins in the 8th, Adam Eaton in the 18th, and Mike Adams in the 20th... snicker snicker) and I ended up with a very solid team.
I don't think this would have worked in a league with all good managers, but with a few reactionary managers I was able to buck a lot of trends to start the new one, a strategy that worked extremely well for me in the middle rounds.