Continuous monitoring of network resources enables to make more-informed resource allocation decisions but incurs overheads. We investigate the trade-off between monitoring costs and benefits of accurate state information for a routing problem. In our approach link delays are modeled by Markov chains or hidden Markov models. The current delay information on a link can be obtained by actively monitoring this link at a fixed cost. At each time slot, the decision maker chooses to monitor a subset of links with the objective of minimizing a linear combination of long-run average delay and monitoring costs. This decision problem is modeled as a Markov decision process whose solution is computed numerically. In addition, in simple settings we prove that immediate monitoring cost and delay minimization leads to a threshold policy on a filter which sums up information from past measurements. The lightweight method as well as the optimal policy are tested on several use-cases. We demonstrate on an overlay of 30 nodes of RIPE Atlas that we obtain delay values close to the performance of the always best path with an extremely low monitoring effort when delays between nodes are modeled with hierarchical Dirichlet process hidden Markov models.