I will be in Greece in Septbember for business. I am planning to take somewhere from 8-10 days vacation once business is done to travel around Greece. However the idea struck me to hop on a flight over to Cairo from Athens and spend about 3 days in Cairo. Is this a good idea? Or should I concentrate on Greece and leave Egypt for a seperate trip?

There is certainly plenty to see and do in Greece. Have you been to any of the islands? Thoroughly recommend Santorini if you can only do one and there is plenty to see on the mainland as well.
A side trip to Egypt is quick and relatively easy from Athens (two-hour flight on Egypt Air).
8-10 days is a great amount of time to spend in Egypt. You can have a few days in Cairo to check out - Islamic Cairo (Khan El Khallili bazaar, Citadel, Alabaster Mosque), Coptic Cairo (Hanging Church) and of course, the Cairo Museum and Pyramids. You can then either independently head south (fly or train) or catch a tour, to Luxor (see Valley of Kings and Valley of Queens, as well as Karnak and Luxor Temples), and cruise to Aswan. Most cruises stop at Edfu and Kom Ombo Temples on the way. Then in Aswan, do a day trip to Abu Simbel, the most spectacular monument in Egypt after the pyramids. You can fly/train back to Cairo (either independently or on tour).
Egypt and Greece are both amazing places to see. You just need to work exactly what sort of holiday experience you are after.
Hope that helps.