React JS - Uncaught TypeError: this.props.data.map is not a function

You need to create an array out of props.data, like so:

data = Array.from(props.data);

then will be able to use data.map() function

The .map function is only available on array.
It looks like data isn't in the format you are expecting it to be (it is {} but you are expecting []).

this.setState({data: data});

should be

this.setState({data: data.conversations});

Check what type "data" is being set to, and make sure that it is an array.

Modified code with a few recommendations (propType validation and clearInterval):

var converter = new Showdown.converter();

var Conversation = React.createClass({
  render: function() {
    var rawMarkup = converter.makeHtml(this.props.children.toString());
    return (
      <div className="conversation panel panel-default">
        <div className="panel-heading">
          <h3 className="panel-title">
            {this.props.id}
            {this.props.last_message_snippet}
            {this.props.other_user_id}
          </h3>
        </div>
        <div className="panel-body">
          <span dangerouslySetInnerHTML={{__html: rawMarkup}} />
        </div>
      </div>
    );
  }
});

var ConversationList = React.createClass({
 // Make sure this.props.data is an array
  propTypes: {
    data: React.PropTypes.array.isRequired
  },
  render: function() {

    window.foo            = this.props.data;
    var conversationNodes = this.props.data.map(function(conversation, index) {

      return (
        <Conversation id={conversation.id} key={index}>
          last_message_snippet={conversation.last_message_snippet}
          other_user_id={conversation.other_user_id}
        </Conversation>
      );
    });

    return (
      <div className="conversationList">
        {conversationNodes}
      </div>
    );
  }
});

var ConversationBox = React.createClass({
  loadConversationsFromServer: function() {
    return $.ajax({
      url: this.props.url,
      dataType: 'json',
      success: function(data) {
        this.setState({data: data.conversations});
      }.bind(this),
      error: function(xhr, status, err) {
        console.error(this.props.url, status, err.toString());
      }.bind(this)
    });
  },
  getInitialState: function() {
    return {data: []};
  },

 /* Taken from 
    https://facebook.github.io/react/docs/reusable-components.html#mixins
    clears all intervals after component is unmounted
  */
  componentWillMount: function() {
    this.intervals = [];
  },
  setInterval: function() {
    this.intervals.push(setInterval.apply(null, arguments));
  },
  componentWillUnmount: function() {
    this.intervals.map(clearInterval);
  },

  componentDidMount: function() {
    this.loadConversationsFromServer();
    this.setInterval(this.loadConversationsFromServer, this.props.pollInterval);
  },
  render: function() {
    return (
      <div className="conversationBox">
        <h1>Conversations</h1>
        <ConversationList data={this.state.data} />
      </div>
    );
  }
});

$(document).on("page:change", function() {
  var $content = $("#content");
  if ($content.length > 0) {
    React.render(
      <ConversationBox url="/conversations.json" pollInterval={20000} />,
      document.getElementById('content')
    );
  }
})