Requesting removal of stored website data from search engines

Greetings fellow developers,

I would like to ask for help regarding the following problem: Is there a way to request removal of stored website data from search engines? Most of the links that show up when searching my domain are old and non-existent.

What I've found from personal research regarding question/problem:

From my personal research I have found that removal requests can be made individually to the well-known search engines such as Google, Yahoo and Bing, but this is not what I am looking for, since I am well-aware that it would take a lot of time for the requests to be processed and the removal of the data to be done. Also, I wasn't able to find this "removal-request" webpage for the other (most) search engines.

To be more precise/clear...

... I want to request this website-data-removal to all (most) search engines at once, so that when I upload my new website (to the same domain), working and functional links (URLs) would be displayed. Can this be anyhow achieved and, if so, how? Also, how much time would it take for this removal to be finished?

Hope my question is clear enough, and any answer/help would be very much appreciated.