In addition to the option suggested by @MXDNS, you can also use smtp2go.com, which has a free tier. I think it permits the sending of up to 1000 emails per month on the free tier, which I never come close to hitting. All of the benefits of an external SMTP service with none of the cost!
The caveat is that you have to make sure that your outgoing mail is absolutely pristine to avoid them booting you from the service.
I had similar issues as you and it came from the incorrect use of aliases to forward mail.
In addition, I had problems getting SRS to work properly for mail that was forwarded using sieve filters in roundcube. I ended up using postfix generic maps to handle the forwarding and that seems to keep everyone happy.