As of .NET 4.5, the SmtpClient class implements an async awaitable method, SendMailAsync. As a result, to send email asynchronously is as following:

public async Task SendEmail(string toEmailAddress, string emailSubject, string emailMessage)
{
    var message = new MailMessage();
    message.To.Add(toEmailAddress);

    message.Subject = emailSubject;
    message.Body = emailMessage;

    using (var smtpClient = new SmtpClient())
    {
        await smtpClient.SendMailAsync(message);
    }
}