If you want (or have to) to use JavaScriptSerializer the code could look like following:
using System;
using System.Collections.Generic;
using System.Web.Script.Serialization;
namespace JsonSer {
    public class MyTag {
        public string tagName { get; set; }
        public string tagType { get; set; }
        public long tagCreateDate { get; set; }
        public int tagOffset { get; set; }
    }
    public enum MyMedia {
        Diskette,
        UsbStick,
        Disk,
        Internet
    }
    public class MyConversation {
        public string conversationId { get; set; }
        public string state { get; set; }
        public string conversationType { get; set; }
        public MyMedia mediaType { get; set; }
        public long startDate { get; set; }
        public int duration { get; set; }
        public List<MyTag> tags { get; set; }
    }
    public class MyConversations {
        public List<MyConversation> conversations { get; set; }
    }
    public class MyData {
        public string responseCode { get; set; }
        public string responseMessage { get; set; }
        public MyConversations responseBody { get; set; }
    }
    class Program {
        static void Main (string[] args) {
            MyData data = new MyData () {
                responseCode = "200",
                responseMessage = "OK",
                responseBody = new MyConversations () {
                    conversations = new List<MyConversation> () {
                         new MyConversation() {
                             conversationId = "conversation1",
                             state = "state1",
                             conversationType = "per JSON",
                             mediaType = MyMedia.Internet,
                             startDate = DateTime.Now.Ticks,
                             duration = 12345,
                             tags = new List<MyTag>() {
                                 new MyTag() {
                                     tagName = "tagName1",
                                     tagType = "tagType1",
                                     tagCreateDate = DateTime.Now.Ticks,
                                     tagOffset = 1
                                 }
                             }
                         }
                     }
                }
            };
            Console.WriteLine ("The original data has responseCode={0}", data.responseMessage);
            JavaScriptSerializer serializer = new JavaScriptSerializer ();
            string json = serializer.Serialize (data);
            Console.WriteLine ("Data serialized with respect of JavaScriptSerializer:");
            Console.WriteLine (json);
            MyData d = (MyData)serializer.Deserialize<MyData> (json);
            Console.WriteLine ("After deserialization responseCode={0}", d.responseMessage);
        }
    }
}
the corresponding JSON data will be look like
{
    "responseCode": "200",
    "responseMessage": "OK",
    "responseBody": {
        "conversations": [
            {
                "conversationId": "conversation1",
                "state": "state1",
                "conversationType": "per JSON",
                "mediaType": 3,
                "startDate": 634207605160873419,
                "duration": 12345,
                "tags": [
                    {
                        "tagName": "tagName1",
                        "tagType": "tagType1",
                        "tagCreateDate": 634207605160883420,
                        "tagOffset": 1
                    }
                ]
            }
        ]
    }
}
You can easy modify the code if you decide to use DataContractJsonSerializer.