Here's a somewhat different answer to those already given.

Associated to any homotopy pullback square, there's a long exact sequence of homotopy groups often called the Mayer-Vietoris sequence. It comes from weaving together the long exact sequences for, say, the two vertical maps in the square, which have homotopy equivalent homotopy fibers. (This weaving is a standard homological algebra exercise, and appears somewhere in Hatcher's book...)

Now, to build the Mayer-Vietoris sequence in cohomology for a CW complex X written as a union of subcomplexes $X = A\cup B$, just note that the homotopy pushout square formed by $A\cap B$, A, B, and X becomes a homotopy pullback square after applying Map(-, K(G, n)), where G is the coefficient group you're using. The Mayer-Vietoris homotopy sequence is now precisely the M-V sequence in cohomology.

(Annoyingly, for a fixed value of n this only gives you some of the sequence.)

It would be interesting to see a variant of this for homology, maybe using the infinite symmetric product? I suppose the place to look would be the book by Aguilar-Gitler-Prieto, where homology is introduced entirely in terms of symmetric products. The relevant bit seems to be missing from the Google preview.

Here's an answer somewhat different from those already given.

Associated to any homotopy pullback square, there's a long exact sequence of homotopy groups often called the Mayer-Vietoris sequence. It comes from weaving together the long exact sequences for, say, the two vertical maps in the square, which have homotopy equivalent homotopy fibers. (This weaving is a standard homological algebra exercise, and appears somewhere in Hatcher's book...)

Now, to build the Mayer-Vietoris sequence in cohomology for a CW complex X written as a union of subcomplexes $X = A\cup B$, just note that the homotopy pushout square formed by $A\cap B$, A, B, and X becomes a homotopy pullback square after applying Map(-, K(G, n)), where G is the coefficient group you're using. The Mayer-Vietoris homotopy sequence is now precisely the M-V sequence in cohomology.

(Annoyingly, for a fixed value of n this only gives you some of the sequence.)

It would be interesting to see a variant of this for homology, maybe using the infinite symmetric product? I suppose the place to look would be the book by Aguilar-Gitler-Prieto, where homology is introduced entirely in terms of symmetric products. The relevant bit seems to be missing from the Google preview.