What you have to do is to create an identical email account ("MyName") at the new host.
Take note that during propagation some emails will end up at your old provider. There are number of solutions to retrive those emails. One would be to just forward all emails to an email address at another provider (maybe a free one), and setup this prior to changing nameservers (in order to point your domain to the new host).
Another one would be to access web mail at your old using that server's IP -- if possible.