I was able to get this converter to work.
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Reflection;
using Newtonsoft.Json;
using Newtonsoft.Json.Converters;
public class CustomDictionaryConverter : JsonConverter
{
    public override bool CanConvert(Type objectType)
    {
        return (typeof(IDictionary).IsAssignableFrom(objectType) || 
                TypeImplementsGenericInterface(objectType, typeof(IDictionary<,>)));
    }
    private static bool TypeImplementsGenericInterface(Type concreteType, Type interfaceType)
    {
        return concreteType.GetInterfaces()
               .Any(i => i.IsGenericType && i.GetGenericTypeDefinition() == interfaceType);
    }
    public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
    {
        Type type = value.GetType();
        IEnumerable keys = (IEnumerable)type.GetProperty("Keys").GetValue(value, null);
        IEnumerable values = (IEnumerable)type.GetProperty("Values").GetValue(value, null);
        IEnumerator valueEnumerator = values.GetEnumerator();
        writer.WriteStartArray();
        foreach (object key in keys)
        {
            valueEnumerator.MoveNext();
            writer.WriteStartObject();
            writer.WritePropertyName("key");
            writer.WriteValue(key);
            writer.WritePropertyName("value");
            serializer.Serialize(writer, valueEnumerator.Current);
            writer.WriteEndObject();
        }
        writer.WriteEndArray();
    }
    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        throw new NotImplementedException();
    }
}
Here is an example of using the converter:
IDictionary<string, int> dict = new Dictionary<string, int>();
dict.Add("some key", 1);
dict.Add("another key", 5);
string json = JsonConvert.SerializeObject(dict, new CustomDictionaryConverter());
Console.WriteLine(json);
And here is the output of the above:
[{"key":"some key","value":1},{"key":"another key","value":5}]