It does not work that way in Russian. In modern Russian objects of most negative verbs generally stay Accusative (иметь is the major exception) while some verbs taking abstract objects like Genitive more. It is technically possible to use a Genitive object with many other negated verbs. However, this sounds stronger and often—old-fashioned. In some cases Genitive is associated with excluding any objects of the kind whereas Accusative suggests a more definite object.

A century ago most verbs would, probably, use Genitive in negative sentences.

The issue of choosing the correct case here is very complicated and is way beyond what this course can teach you. A beginner should know that не иметь always uses Genitive and that many abstract objects used with verbs of quite abstract meaning will also use Genitive when negated. Set expressions should be memorized:

• не обращать внимания = to not pay any attention
• не вызывать сомнений = to cause no doubts
• не видеть смысла = to not see any point

Verbs of visual and auditory perception may also switch to Genitive (the interpretation being "we trust what we see and hear, so if we do not see or here something, it probably is not there"). This actually extends to subjects of perception predicates ("is visible", "is audible"):

• Я не видел дома. = I did not see the house.
• Я не слышал шума. = I did not hear any noise.
• В соседней комнате музыки не было слышно. = The music "was not audible" in the adjacent room.
• Тебя отсюда не видно. = You are "not visible" from here.