Techniques are described for managing customer-specified routing policies for network-accessible computing resources. In some situations, the customer-specified routing policies may be based at least in part on DNS (“Domain Name System”) information specified by a customer, such as if the customer specifies one or more target destinations to use with an indicated DNS domain name that are different from the destination IP address(es) provided for that DNS domain name by DNS servers—if so, the managing of such a DNS-based routing policy for that customer may include identifying when network-accessible computing resources provided to the customer send electronic communications to that DNS domain name, and causing those electronic communications to be redirected to the customer-specified target destination(s). Such customer-specified target destinations may include, in different situations, final destinations, intermediate destinations, etc., as well as identify particular routes.