Is there a good way to figure out whether my pet is running a temperature?

Use a thermometer.

The normal body temperature of a cat or dog at rest is 99 to 101ºF. If your pet is acting sluggish or depressed, is reluctant to eat or move around, is panting or breathing rapidly, or feels warm to the touch, it is exhibiting symptoms of a fever. The most accurate method of determining whether an animal has a fever is to use a rectal thermometer.

As with taking the temperature of an infant, the thermometer should be shaken, lubricated with petroleum jelly, and inserted about one-and-a-half inches into the rectum. It is helpful to have someone hold and distract your pet while its temperature is being taken.

If the reading is above 102.5ºF, you should contact your veterinarian. A temperature of 105ºF or more will require an emergency visit to a veterinarian. If the pet holds a temperature above 106ºF for very long, it could suffer serious adverse effects, including permanent brain damage and death.