1999 Cherokee intermittent misfire

Problem: My 1999 jeep cherokee sport (94,000 mi, 4.0L - 6) starts fine every time. After 15-20 seconds it starts missing. If you put it under load it "backfires through the carb" (at least that's what it sounds like). It misses for about 15-20 seconds. Then the problem goes away. It's the same every time it happens. There is never a check engine light. Doesn't seem to matter if engine is warm or cold. Sometimes it will do it after only a two minute stop. But it doesn't do it every time.

My dealer said they saw the problem but could not find what was causing it, since it goes away so quickly. They said they would rather I drove it and when it got worse, they'd try to fix it. They said the only other option was to "easter egg" it. They would keep replacing parts until it got fixed. Well, it has now been doing it for about a year. No better. No worse.

Could use some help. At the very least, if I start replacing parts myself, which ones and in what order should I replace them.