Everything will be given in the case where we are computing f(n,i,j) for 2*n+i=2*n+j=0 mod 4. 

The rules for numbering the cells: the face cells are numbered with three indices 
(n,i,j), of which n is actually redundant. i and j are always even and congruent mod
 4. n is always the unique value such that 6*n+j=4, 8 or 12. (The specific one that
 occurs is the one which is congruent to j mod 6.) The resulting n values go 3, 2, 1,
 2, 1, 0, 1, 0, -1, 0, -1, -2. A face corresponds to a square if 6*n+j=4 or 12 and a
 hexagon if 6*n+j=8. (In other words, squares come from j=0 or 4 mod 6, hexagons come
 from j=2 mod 6.) The (i,j) locations of the cells form a square grid at 45 degrees 
from the axes where i is constant on columns and j on rows in the simple manner below:
   
     *         *         *
    / \       / \       / \ 
   /   \     /   \     /   \
  /     \   /     \   /     \
 /       \ /       \ /       \ 
* 2,-4,-4 * 2,0,-4  * 2,4,-4  *
 \       / \       / \       /
  \     /   \     /   \     /
   \   /     \   /     \   /
    \ /       \ /       \ /
     * 1,-2,-2 * 1,2,-2  *
    / \       / \       / \
   /   \     /   \     /   \
  /     \   /     \   /     \
 /       \ /       \ /       \
|         |         |         |
|  2,-4,0 |  2,0,0  |  2,4,0  |
|         |         |         |
 \       / \       / \       /
  \     /   \     /   \     /
   \   /     \   /     \   /
    \ /       \ /       \ /
     * 1,-2,2  *  1,2,2  *
    / \       / \       / \
   /   \     /   \     /   \
  /     \   /     \   /     \
 /       \ /       \ /       \
* 0,-4,4  *  0,0,4  *  0,4,4  *
 \       / \       / \       /
  \     /   \     /   \     /
   \   /     \   /     \   /
    \ /       \ /       \ /
     |         |         |
     | 1,-2,6  |  1,2,6  |
     |         |         |
    / \       / \       / \
   /   \     /   \     /   \
  /     \   /     \   /     \ 
 /       \ /       \ /       \
* 0,-4,8  *  0,0,8  *  0,-4,8 *
 \       / \       / \       /
  \     /   \     /   \     /
   \   /     \   /     \   /
    \ /       \ /       \ /
     *         *         *

      
The edges around a face are linear functions of n,i and j, depending on the value of
 j mod 6. (Of course, j determines n, but the formulas are less elegant if written 
only in terms of n and i.) The second coordinate only depends on j and n (is constant
 in rows.)

j=0 mod 6

                  *
                 / \
2n+i-1,2n+j-1   /   \  -2n+i-1,2n+j-1 
               /     \
              * n,i,j *
               \     /
2n+i-1,-2n+j+1  \   /  -2n+i-1,-2n+j+1
                 \ /
                  *

j=2 mod 6

                  *
                 / \
-2n+i-3,-2n+j-3 /   \  2n+i+3,-2n+j-3
               /     \
              *       *
              | n,i,j |
              *       *
               \     /
2n+i-1,-2n+j+1  \   /  -2n+i+1,-2n+j+1
                 \ /
                  *

j=4 mod 6

                  *
                 / \
-2n+i-3,2n+j-3  /   \  2n+i+3,2n+j-2
               /     \
              * n,i,j *
               \     /
-2n+i-3,-2n+j+1 \   /  2n+i+3,-2n+j+1
                 \ /
                  *

Now, we must describe how to find the graph assosciated to f(N,I,J) for a particular 
(N,I,J). The faces of this graph can either be found by expanding out the rule that 
(N,I,J) depends on (N-1, I +/- 2, J +/- 2) and (N-2,I,J) for all for pairs of choices
 of + and -. Algebraically, these are the sets of (n,i,j) such that the four 
inequalities 2n-j<=2N-J, 2n+j<=2N+J, 2n-i<=2N-I, 2n+i<=2N+I are satisfied. To draw 
the graph, draw the edges surrounding the faces for which strict inequality holds in
 all cases. For a sample graph with all edges labeled see www.fas.harvard.edu/~speyer/somosgraph 
and for many of these graphs without labels see 
www.fas.harvard.edu/~speyer/somosgraph2.html.

Finally, we must describe how each monomial corresponds to a matching of a graph. 
An edge variable appears in a monomial iff that matching contains that edge. A face 
variable for a square face occurs wih exponent 1-#(edges adjacent to face) and a 
face variable of a hexagon occurs with exponent 2-#(edges adjacent to face).

I should add that, the above scheme will describe Aztec diamonds if we replace the 
formula for n in terms of j by n=0 or -1, according to j mod 4. (The i and j 
coordinates are doubled from their usual indexing and I think the edge weightings 
are halved and slightly shifted, I haven't quite gotten it.) The rules for how to 
label the edges around a face then correspond to j mod 4, not mod 6. These two cases 
can be synthesized by saying that the first, second and third rules (j=0, 2, 4 mod 6 
in the original Somos-3 problem) should be used when n is (a) greater than n for all 
its neighbors, (b) greater than the n for two and less than n for the other two or 
(c) less then all its neighbors respectively.