Contact the sites from which you would like links and ask for one.
If you are building a site for somebody else they should are the subject matter expert and should advise you which are the most appropriate sites to approach. If you are building it on your own account then you are the SME and should know already.
I built a site for a regional arts organisation once - took two lines of approach:
1. Identify national organisations and media sites that provided links.
2. Identify local organisations and artists with which the organisation was associated, linked to them and requested return links
It was highly successful and spawned links from many other sites. The most productive links were not from these sites rather than the ones that I had approached direct