I don't think that there is one specific Torah strategy for combating greed, and I think you may need to experiment with a few to see which is most effective for you. Here are some strategies that may be helpful:
1. Gratitude - The more one expresses and feels gratitude for what one already has, the less that one will desire more. When you eat, say a blessing beforehand, and focus a little on being grateful for what you have. This will make you happier, less inclined to look at what others have, and less compelled to seek more. During the silent prayer, when we say the blessing of thanks, take some time to meditate on the aspects of your life for which you are grateful - family, community, health, etc. (Kuzari 3:11)
2. Humility - Try to see yourself as a recipient of God's kindness, and the kindness of other people, not as one who deserves or is entitled by rights to what he has. A little self-criticism and understanding that maybe we are not so worthy goes a long way to helping us cope with greed and envy. (Letter of Nachmanides)
3. Channel your ambition and energy - and greed (!) into positive areas. Be greedy about Torah wisdom, try to accumulate as much as possible. Look with envy upon good character traits in others and try to acquire them yourself. (Shem MiShmuel, Parshat Korach)
4. When you perform a mitzvah (and do almost any other activity) try to focus on that activity completely. Immerse yourself in the experience and you will find it a more satisfying and meaningful experience, one that you will want to return to, one that will give you joy. (Nachmanides, Critique of Book of Commandments, Positive Mitzvah 5)
5. Give - involve yourself in giving to others. Either by volunteering, giving money or helping out professionally. When you give to others you become more appreciative of your own good fortune, you become less inclined to be jealous, you will create friendships, and you will be become less of a "taker" and more of a "giver". (Rabbi Eliyahu Dessler, Michtav MeEliyahu, Kuntres HaChessed)
6. Contemplate and study ideas about Divine Providence and trust in God (E.g. Duties of the Heart by Rabeinu Bachya, Section on Trust). Try to come to an understanding that God gives what you need and precisely that which is appropriate to you in your situation. (Ibn Ezra, Exodus 20:14)
