I have a Kenmore furnace (I assume it's made by Goodman), model GMP100-4, 1993. The initial problem was that the furnace only operated intermittently, it would run most of the day and then stop. I had to turn the furnace off to get it to operate again. After a few days of this this the furnace only stayed on for brief periods. I changed the Flame sensor and the HSI. It worked for another week. I then updated the thermostat to a Honeywell RTH6300B. That worked for about another week. Now I can't turn the furnace on at all. While at the furnace when I turn the power off and then back on the inducer blower motor comes on for about 5 sec. then stops. The LED on the control board blinks 3 times continuously. The vent and vacuum hose appear to be clear. I used my mouth to apply a low vacuum on the pressure switch and my VOM indicated zero ohms proving that the switch contacts were closed.
What do you think the problem is? What other components should I check and how do I check them?

Three blinks on Goodman GMP100's usually is caused by the vacuum pressure switch. Have you check your flue and vent pipes to be sure they are clean and don't have any blockage? Please be careful and don't try any repairs that will put you and your family in danger.