No grounded outelsts

Get an outlet tester and try replacing one of the outlets with a new 3-prong one. (Be sure to kill the power first.) You may find that you are safely grounded without having to upgrade the wiring. Worked in my 1948 house.

An easier way to get the same answer would be to use a cheater plug in the existing outlet but ground the cheater plug to the center screw in the outlet cover. All cheater plugs will have either a metal tab with a hole in the center of it or a green ground wire with a connection on the end. Those are designed to be connected to the center screw on the outlet plate, creating a connection to the outlet box, itself, through the outlet cover screw. Testing for that with an outlet tester or multimeter will tell you.

Some outlet boxes are grounded by the metal coax through which the wires run but some older ones are not grounded at all. If you're lucky, your boxes will be grounded and you can either use the grounded cheater plugs or replace the outlets with 3-prong ones as Drubin suggested.

Replace one outlet with a grounded duplex outlet. The use one of those inexpensive electric outlet testers with three lights (home depot/ace/lowes/....) and see if the outlet is grounded. If it isn't, the lights will show you.

why guess when you can actually find out for less than $10.