When I read the Greek New Testament however, the genitives stand out, and I find myself analyzing what they mean (especially thanks to Wallace's extensive section on Genitives in his grammar).
For example, today I was reading:
John 15:26 "But when the Helper comes, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will bear witness about me.
I have read the phrase 'Spirit of truth' so many times in English, and never thought too much about it. But today, the Greek was begging me to identify what kind of genitive it was. I think that the most obvious category for this is the genitive of product. In other words, the Spirit of truth, is the Spirit that produces truth. The context indicates this. The Holy Spirit will produce truth and witness about Jesus. I found it so helpful to identify what "of truth" specifically meant.
It helped me to mediate on the fact that the Holy Spirit produces truth. It led me to pray that the Holy Spirit would enable to understand the word as I prepare tomorrow's sermon, also for truth to be given to the congregation when they hear the sermon, and for truth to be produced on our council estate, so that people would come to Christ.
It also makes me consider the possibility, or helping non-Greek readers to study the Bible. What if we taught people categories of the word 'of' and to stop when they see the word 'of' and consider what type of genitive it is? I haven't given this enough thought to know if it would work, but its something to think about.