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The short answer is yes most definitely you should distribute articles to more than one directory.  Let me clarify that answer though.  You should not distribute the same article to more than one directory if you want maximum benefit from the links.  If you want maximum benefit then you should be distributing a unique version of each article to many different directories.

Google does not like duplicate content.  That was part of the reason they came out with the Google Panda update.  Websites that publish too much of the same exact material that is published at lots of other places are considered to be low quality websites.  So any site that is going to accept an article from you that you had already published elsewhere is going to be considered low quality in the eyes of Google.  Getting a link from that site might not harm you, but it also is not likely to help you all that much.  There are better ways of doing this.

Is it even important to get links from that many different sites?

Yes absolutely it is.  It is important if you want to make your rankings move higher.  You want links from as many different places as you can get them.  The more unique websites you have linking back to you, the better.

What is the best way to do this then?

You will have to decide that for yourself.  If you are a white hat article marketer then you are going to need to write and publish one unique article for every new site that you publish on.

If you don’t care to follow the white hat rules then you could get a lot more links by using an article spinner or a service that combines article spinning and distribution.  There are lots of them who do that.