You can declare a custom contract resolver which indicates which properties to ignore.  Here's a general-purpose "ignorable", based on the answer I found here:
/// <summary>
/// Special JsonConvert resolver that allows you to ignore properties.  See https://stackoverflow.com/a/13588192/1037948
/// </summary>
public class IgnorableSerializerContractResolver : DefaultContractResolver {
    protected readonly Dictionary<Type, HashSet<string>> Ignores;
    public IgnorableSerializerContractResolver() {
        this.Ignores = new Dictionary<Type, HashSet<string>>();
    }
    /// <summary>
    /// Explicitly ignore the given property(s) for the given type
    /// </summary>
    /// <param name="type"></param>
    /// <param name="propertyName">one or more properties to ignore.  Leave empty to ignore the type entirely.</param>
    public void Ignore(Type type, params string[] propertyName) {
        // start bucket if DNE
        if (!this.Ignores.ContainsKey(type)) this.Ignores[type] = new HashSet<string>();
        foreach (var prop in propertyName) {
            this.Ignores[type].Add(prop);
        }
    }
    /// <summary>
    /// Is the given property for the given type ignored?
    /// </summary>
    /// <param name="type"></param>
    /// <param name="propertyName"></param>
    /// <returns></returns>
    public bool IsIgnored(Type type, string propertyName) {
        if (!this.Ignores.ContainsKey(type)) return false;
        // if no properties provided, ignore the type entirely
        if (this.Ignores[type].Count == 0) return true;
        return this.Ignores[type].Contains(propertyName);
    }
    /// <summary>
    /// The decision logic goes here
    /// </summary>
    /// <param name="member"></param>
    /// <param name="memberSerialization"></param>
    /// <returns></returns>
    protected override JsonProperty CreateProperty(MemberInfo member, MemberSerialization memberSerialization) {
        JsonProperty property = base.CreateProperty(member, memberSerialization);
        if (this.IsIgnored(property.DeclaringType, property.PropertyName)) {
            property.ShouldSerialize = instance => { return false; };
        }
        return property;
    }
}
And usage:
var jsonResolver = new IgnorableSerializerContractResolver();
// ignore single property
jsonResolver.Ignore(typeof(Company), "WebSites");
// ignore single datatype
jsonResolver.Ignore(typeof(System.Data.Objects.DataClasses.EntityObject));
var jsonSettings = new JsonSerializerSettings() { ReferenceLoopHandling = ReferenceLoopHandling.Ignore, ContractResolver = jsonResolver };