It ranges from easy to nearly impossible. Without knowing what kind of rig you have (GVWR, motor, type of vehicle you're towing) and which mountains you are talking about, it's pretty hard to give you an answer. I drive a 38 1/2' class A (20,000 GVWR) with a Ford 460 gas engine and pull a Dodge Stealth with a curb weight of 3800. Most mountains don't give me much trouble but a trip this spring thru Tonto National Forest with some 6 to 6 1/2% grades was a strain on both the engine and the brakes. It's not a trip would make again in the MH (Especially trailing a toad), but others with their gazillion hp. diesels may have no problems ;D
If you provide some details, I'm sure others can advise you better and even suggest certain routes to avoid altogether and times of year to avoid others.
In 2003 I pulled a 30 foot Fleetwood travel trailer all over the Rockies with a Dodge Ram 2500 gasser. It was not easy but not impossible. I had numerous occasions of overheating. In most cases I could find a spot to pull over and wait for the engine to cool. A few times, maybe 3 or 4 over a ten week period, I could find no shoulder and had to climb to the top. I downshifted, getting the rpms up, and went as slow as I could but still keep momentum going, and prayed all the way to the top.
If I had it to do over again, I would still do it. But there was risk involved. the big difference to back east is not the steepness, but the length of the inclines. I have since added a transmission cooler, but have given the van and trailer to my son and upgraded to an MH.
You need to be concerned about braking on the downgrades as well as pulling the long upgrades. Watch for overheating the engine on the way up (watch the gauge) and use lower gears to help slow on the way down to assist the brakes. Stopping along the road to allow cooling is wise, especially downhill where you don't have a gauge to tell you what is happening to the brakes.
With caution, you should manage it OK. Just how careful you need to be depends on the details of your weight, power train, etc. that Karl asked about.