I found the answer. I used Postman to do a GET on a BuildType that already had a Trigger configured:
http://teamcity:8111/httpAuth/app/rest/buildTypes/id20_mybuild
In the response you get the full config, including the XML for the Trigger. So, using TeamCitySharp, creating a trigger would look like this:
string xml = @"<trigger id=""TRIGGER_1"" type=""buildDependencyTrigger"">
<properties>
<property name=""dependsOn"" value=""id31_MyProject_SomeBuild""/>
</properties>
</trigger>";
client.BuildConfigs.PostRawBuildTrigger(BuildTypeLocator.WithId("NewBuild"), xml);