The emergence of Linked Data on the WWW has spawned research interest in an online execution of declarative queries over this data. A particularly interesting approach is traversal-based query execution which fetches data by traversing data links and, thus, is able to make use of up-to-date data from initially unknown data sources. The downside of this approach is the delay before the query engine completes a query execution. In this paper, we address this problem by proposing an approach to return as many elements of the result set as soon as possible. The basis of this approach is a traversal strategy that aims to fetch result-relevant data as early as possible. The challenge for such a strategy is that the query engine does not know a priori which of the data sources that will be discovered during the query execution contain result-relevant data. We introduce 16 different traversal approaches and experimentally study their impact on response times. Our experiments show that some of the approaches can achieve significant improvements over the baseline of looking up URIs on a first-come, first-served basis. Additionally, we verify the importance of these approaches by showing that typical query optimization techniques that focus solely on the process of constructing the query result cannot have any significant impact on the response times of traversal-based query executions.