I want a dense green plant that will grow straight up and thin, so as to not be too proud from the house.?

Which one should I use?

Answers:
Italian Cypress will grow pretty high - mine is 30 feet and still growing - and only about 4 or 5 feet in diameter

Other answers:
Juniper bushes can be trained to grow tall, you just have to trim them properly each year.
Juniper bushes can be trained to grow tall, you just have to trim them properly each year.
more than a especies (don´t know where are you living in) it is important the way you grow the fence. if you put many many plants, they will compite, and grow up. but you must cut the plants in order to the ligth to get all over the plants. I mean, the bottom of the plant MUST be wider than the top. if you don´t do that, the plant itself will create a shadow over him, and the lower leaves will fall, because of lack of light. in few words: the more plants you put, the fastest the fence will be completed, and watch the way the plant grow
Try Silver Oak. Or the himalayan Deodhar.
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