  We show that the word problem for any 3-manifold group is solvable in time
$O(n\log^3 n)$. Our main contribution is the proof that the word problem for
admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in
$O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds.
Similar methods also give that the word problem for free products can be solved
"almost as quickly" as the word problem in the factors.
