Try programming it like this:

Press and hold a button on the remote and then press and immediately release the smart/learn button. If this doesn't work you may have a bad logic board but I am wondering why the original board was replaced.

Make sure you do not have a problem with the wall control or the wiring to the control. You can do this by temporarily removing the wire from terminal #1 on the rear of the motor unit.

Check the smart/learn LED for a flashing code. It will sometimes help diagnose a problem. It will flash a number of times, pause and then repeat.

Thanks for you help don_1
I used the press/hold routine you suggested but is did not work. The panel light stays on for about 30 seconds with no results.
I replaced the logic board in an attempt to spend $65 rather than pay a repairman $175. Replacing the board was also the recommendation of the Sears person answering my call for assistance.
I can check the wall control or wiring to the control by removing the wire from terminal
#1 of the motor unit if you can describe which is terminal #1.
The motor has a red, white and blue wire with the red and blue probably from the capacitor. There is also a 4-wire harness yellow/white, purple, and two grays - one of which is smaller than the other.
Can get two responses from the SRT LED . Pressing and hold or press and release and the LED will go off within 30 seconds. Another time is after press/hold a remote button, press/hold wall light button, and press/hold the wall door opener button. The LED flashes 3 times and will repeat that if the buttons are continued to be pressed.

Terminal #1 is on the rear of the motor unit. It should have a white wire with a red stripe going to it. You may have to remove a light lens to get to it. The capacitor is not going to be the problem. If the LED was flashing a code it would do it without you pressing any buttons. What was the problem before you replaced the logic board?

All of the covers of the unit have been removed. The motor unit "End panel" is on the right side of the unit (not at the rear of the unit.) In the entire opener assembly there is but one red striped white wire that runs from the opener assembly at the ceiling up through the ceiling to the wall control. I cannot confirm that it goes to the wall control since I am having trouble removing the cover from the wall control to find the wire. On the right-sided end panel the red stripe goes to a terminal labeled with 1 and the white wire goes to the 2 terminal. (There is another terminal labeled 3 that has a couple of white wires to it.)
The problem which led me to replace the logic board is that the remote controls and Keyless entry unit would not work. I could not get them programmed after numerous attempts using different procedures. Thus I assumed that the logic board was bad since it involves programming remotes.

That red striped wire is the one I was referring to. You seem to have the same problem with the new logic board as you had before. Now there is always the possibility that you got a bad logic board. It happens. Are your remotes and keypad the units that came with the opener and did they work before? Did the opener just quit or was there another problem? See if Sears will exchange the board for you.

I removed the red striped wire from terminal #1 and no change with anything. Without it the door opens/closes and no remote programming. Put it back on the unit and tried programming - nothing. Remotes and keyless entry pad are original - 1998. The entry pad usually worked with difficulty (poorly designed requiring too much pressure) - last used approximately 4 months ago. Remotes not used for a year but worked okay then.