Mayer-Vietoris for a triple Let $X$ be a topological space with $X = A_1 \cup A_2 \cup A_3$.  In the case I am interested all of these spaces are manifolds and submanifolds.  Is there something like an exact sequence relating $H_\ast(X)$ with $H_\ast(A_i), H_\ast(A_i \cap A_j), H_\ast(A_1 \cap A_2 \cap A_3)$, and $H_\ast(A_i \cup A_j)$.  In the application that I have in mind I would like to understand $H_\ast(X)$ an dI understand the homology of all of all of the smaller pieces.  
 A: This is not an exact answer to your question, but an indication of a related question: what should be "homotopical excision" for a space which is the union of more than two open sets? 
An answer is given in Section $1$ of  this paper
R. Brown and J.-L. Loday, "Homotopical excision and Hurewicz theorems for $n$-cubes of spaces" Proc. London Math. Soc.   (3) 54 (1987) 176-192. 
There are two ideas explained there. 
Let $\square(n)$ be the category determined by the partial order $\{0,1\}^n$, where $0 < 1$. An $n$-cube of spaces is defined to be a functor $X: \square(n) \to Top$. Such a functor determines an $n$-cube $X^2$ of $n$-cubes of spaces  by the rule 
$$X^2(\alpha)(\beta)= X(\alpha \wedge \beta), \quad  \alpha, \beta \in \square(n).$$
For example, a square of spaces 
$$\begin{matrix}
C & \to & B \\
\downarrow && \downarrow\\
A& \to & Y
\end{matrix}$$
determines a square of squares of spaces
$$ \begin{matrix}
\begin{matrix}C&C\\
C&C
\end{matrix}& \to & \begin{matrix}C&C\\A&A\end{matrix}\\
\downarrow & & \downarrow \\
\begin{matrix}C&B\\C&B\end{matrix}& \to & \begin{matrix}
C&B\\A&Y\end{matrix}
\end{matrix}$$
Further, an $n$-cube of spaces $X$ can be regarded as a map of $(n-1)$-cubes of spaces
$$e(X): \partial^-_n X \to \partial^+_n X.  $$
The usual excision is where a square of spaces is regarded as a map of maps of spaces. 
Now a $3$-cube $X$ of spaces determines a map $e(X)$ of squares of spaces, and so a map $e(X)^2$ of squares of squares of spaces, which then determines a $3$-cube $Y$ of squares of spaces! 
I leave you to write  out this $3$-cube in the case say of $X= U \cup V \cup W$. 
The advantage of this approach in the cited paper was that we used functors on $n$-cubes of spaces, and a van Kampen theorem for these, giving new  connectivity and algebraic  homotopical results in Theorems $4.3$ and $6.1$.  
This suggests that for a homological result in the case $n=3$ you need to use $H_3(X;A,B)$ . 
I'd be interested to hear if this sketch proves useful. 
