I played like a maniac for a couple of months three years ago or so. Lots of it is just practice, the more you play the faster you can fill in the grid.
With the old site, there were lots of puzzles that were essentially the same in structure, just the story and the names were different. I have an elephant's memory, and I soon started recognizing patterns. After a while I knew which clues I had to look at first to see the pattern emerge. And that is exactly the reason why I stopped playing... I could solve many puzzles really fast but it just got boring. So I am happy to see the new site
What I did though was print some of the puzzles and challenge myself to solve them just filling in the top section. It is not necessary to fill in the whole grid to solve a puzzle, it is just helpful because it makes it easier to see things. So in the end it all comes down to practice, I guess.