And like others do, we always check the engine every hour and it is always the same person who checks them. That way you can hear or see something that is different compared to last check. I also have 3 cameras in the ER, which are always visible on a screen on the flybridge.
We also have the IR thermometer, creates a nice log / trend.
Before we set off I will polish the fuel in all the tanks. All the fuel in every tank passes at least 5 times through the filter so that I know for sure the fuel in the tank is clean and no algae are present. From the tank the fuel then goes to the racors and then to the engine fuel filters. That way I know no contaminated fuel is entering the fuel pumps.
I guess that, if you have run your engines a lot in the previous years, you will also know how much oil it normally uses and if it leaks somewhere. That way you can calculate when you should add oil or fluids while underway. Shutting down both engines in a blue blue situation is something I won't do. I will shut down one engine, keep the other one running, cool down the stopped engine a bit, check all the fluids, start her up again and only then will I shut down the second engine.
I know this may be over the top safe, but you can never say never. I remember a colleague during my pilot training. On one of his solos he decided to shut down one engine, just to find out how it was to fly single engine. And surely enough, he could not restart it anymore, so now had to explain why he, as a student shut down one engine in flight. He never expected that engine not to restart anymore, but it did. So that was a good lesson never to assume, because it makes an 'ass out of you and me'.