Well, more like three biggest problems with Jericho.

In the episode Fallout, the radiation from Denver was coming towards Jericho.

The rain would supposedly clear the radiation out of the air.

1. It would take approx. 4 weeks to completely clear the radiation out of the air.
Not really! Fallout radiation decreases by a factor of ten for every seven hours. By the time the
rain reached Jericho the radiation would be no more than normal.

2. The people in the mine would suffocate to death from dust and lack of air or just get lung damage.
What is this based on?
Not true. The mine had a ventilation system.

3. Oh and the EMP blast later in the season wouldn't affect anything that isn't plugged in.

This isn't necessarily true. An EMP is so destructive because it essentially consists of a massively broad band high energy radio pulse that covers the entire spectrum, has a massive amplitude, and lasts for several seconds. The problem is, electrical conductors function as antennas and waveguides, and when exposed to this much energy, can deliver sufficient charge and current to damage certain electronics (especially transistors).

Also, terrorists could have launched one or more ICBMs to cause EMP from high-altitude detonations.

These missiles were likely launched from Chechnya,or another former Soviet province. Muslims extremists have been battling over southern Russia since the Soviet Union broke apart in the early 1990s. Security and reliability at Soviet missile bases and silos are notoriously ineffective or absent entirely.

4. Denver and the Rocky Mountains are not visible from Kansas. The tops of the Rockies are only glimpsed once a traveler is well inside Colorado, at the junction of highway 24 and Interstate 70, near Limon, Colorado. Looking west from Goodland KS, you would not see the Denver mushroom cloud. Probably, you would only notice a sudden brightening of the western sky.
The location of Jericho appears to have been specifically designed to not reference a real location but rather a generic small town. Focusing on geographical inconsistencies is pointless because Jericho isn't a real town to begin with.


5. In a nuclear attack designed to cripple the country, Colorado Springs is a better target than Denver. A small nuclear blast would destroy or seriously damage key military assets: the US Air Force Academy, USAF Space Command HQ at Peterson Air Force Base (also destroying or crippling NORAD, which recently moved to Peterson from Cheyenne Mountain. The Air Force recently disclosed that modern nuclear weapons would literally shatter the mountain above NORAD); and Fort Carson. Satellite control and early warning capabilities at Schriever AFB (in Ellicott CO, east of the Springs) and Denver's Buckley AFB would be severely crippled.
Although the intentions of the terrorists may not have necessarily been to cripple the US military infrastructure. As we've seen, soft targets and civilian centers like NYC or Philadelphia, which hold little to no military value, have also been targeted. As a means of creating confusion and inflicting the maximum amount of economic and political damage the objective would be to strike away from military bases and focus on other targets.