At the service desk you cannot always choose the way notifications are delivered, this may fall into the hands of whoever setup the workflow of your system. There may be no way to stop the emails of ticket notifications and approvals because your IT Director still thinks this is the most efficient way. I don't want to even get started with how this system is for the customer, because it really is a whole article in itself. So at this point change is not an option for you, how can you handle this for yourself?
I suggest setting up a rule so that all tickets route into a specific folder. Ignore the notifications and phone vibrations and work straight from your IT dashboard or ticket view. Of course, if you are mobile, your ITSM solution will need a Smartphone client, otherwise email may be the only option. As for a more collaborative approach, consider the use of an enterprise social tool where your team can posts about what they are working on and people can reply, just don't activate email notifications for the tool.
While I'm a proponent for reducing email use, it's not time to kill email - it's time to evaluate how you are using it and make sure that there is actual substance being exchanged. Consider your fellow workmates and their email habits to get an idea of how to make the best changes.