That's where I went. I found the parts I need pretty inexpensive at Arnold's: http://arnoldservice.com/bryant_carrier_parts.htm, so I just added a limit switch to my order for the 3-wire pilot. I figure I'll install the new parts and see what happens. I guess, if the the cycling continues, I'll be looking for air obstructions.
I just wanted to post an update to say "thanks" for all the help and to let everyone know the final outcome. I replaced the 3-wire pilot and the limit switch, but broke the darn connector off the fusible link in the process. I tried soldering the link connector, but once that was done, the link no longer had continuity. After replacing the link, everything worked like a charm.