In the Watchman Route Problem (WRP) we are given a grid map with obstacles and the task is to (offline) find a (shortest) path through the grid such that all cells in the map can be visually seen by at least one cell on the path. WRP was recently formalized and optimally solved with heuristic search. In this paper we show how the previous optimal methods can be modified (by intelligently pruning away large subtrees) to obtain suboptimal solvers that are much faster than the optimal solver without sacrificing too much the quality of the solution. In particular, we derive bounded suboptimal solvers, suboptimal solvers without bounds and anytime variants. All these algorithms are backed up with experimental evidence that show their benefits compared to existing approaches.