We had, spread over the course of our 8th grade English class (Thanks Ms Wilson), about 500 greek and roman roots to memorize, and weekly quizzes. These were not graded curricula, they were for extra credit because it was the teacher's personal program. No grammar, no conjunctions or conjugations, no sentence construction, just the two biggest veins that PIE has contributed to English nouns and verbs. Rote memorization.
I found I already could guess about 2/3 of them from being a recreational reader, but it helped a good deal even so. With the combination of a few years of Spanish and random etymological crawls through Wikipedia, I'm firmly in the top few percentiles of English vocabulary competence.
That sounds great, and very different from what GP said ("make French/Spanish/etc the second foreign language they learn after Latin&Greek").
Edit: I will still say that Greek has little relevance to common English vocabulary, though it is very relevant to almost every scholarly domain. The same is true to some extent for Latin - as the vast majority of non-scholarly Latin words in English are actually borrowed from French, and have (Old) French spelling and pronunciation, not Latin ones.